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Plate 1 





Author of "Elements of Physics," "Introduction to 
Physical Science," etc. 

Boston, U.S.A., and London 




Entered at Stationers' Hall 

Copyright, 1895 


cr ' 'N3 ^ '^ 


It is now thirteen years since my Elements of Physics was 
published. In the interim many changes in the technical no- 
menclature of Physics and many improvements in methods of 
presentation of portions of the science have been made, and, 
above all, the whole subject of Electricity and Magnetism has 
outgrown its former apparel. Furthermore, the present is 
conceded to be an era of extraordinary scientific activity, and 
the demand for frequent revisions of scientific text-books is 
correspondingly imperative. 

The present volume, however, is a new work, and not a 
mere revision of former works. Much of the material of 
previous works, when suited to present needs, has naturally 
been incorporated into this ; but everything has been carefully 
rewritten and rearranged with reference to its adaptability to 
the requirements of the present day. 

The considerable increase of volume and scope of this work 
over those of its predecessors may require some explanation. 
In this book I have naturally been moved to attempt to meet 
the demands of many highly esteemed critics who have com- 
plained of serious omissions in my former works, and I am 
apprehensive solely lest this volume also may fall short of the 
requirements of many. Then, too, as was suggested above, the 
scientific activity of this age results in successive additions 
both to the theory of Physics and to its application, and thus 
tends to make the ground covered by new text-books con- 
tinually greater. 


But tMs book represents more than this : it represents the 
author's protest against a tendency in some quarters towards 
demanding ^^ smaller books," '^ cheaper books," "primers of sci- 
ence," — a protest based on the conviction that true education 
does not consist in the acquisition of the fewest possible facts 
about any subject. Education, in Physics, implies the presen- 
tation of the great truths of that science in their unmutilated 
form, the indication of their relations to one another, and the 
furnishing the student an o'pportunity of observing and exercis- 
ing the logical processes that have led to the discovery of those 
truths. Any text-book that aims to introduce the student 
to a study of such importance and such inexhaustible possibili- 
ties should not lose sight of this truth and encourage mere 
dilettanteism. In particular there is needed a store of illustra- 
tive matter, of concrete applications of general principles, 
sufficient to make clear those principles and to indicate the 
inductive processes by which they have been reached and the 
deductions to which they lead. 

Meagre information results in hazy comprehension, and 
consequently provokes but meagre interest. Full and varied 
treatment, on the contrary, by presenting different points of 
view, clears the conceptions and thus provokes interest, and 
allures to continued study. All things considered, too much 
in a text-book is far preferable to too little. 

In these considerations may be found a partial explanation 
of the size of this volume : a work which aims to afford the 
possibility and the ijicentive to more than a superficial know- 
ledge of the subject. 

The work contains two courses — one which is termed a high 
school course, and the other an advanced course. The former Is 
printed in larger type ; the latter comprises the former and 
additional matter j)rinted in smaller type, which is indented 
about one-fourth of an inch at the left margin of the page. 
The former embraces a full course for those high schools and 


academies which are able to do a fairly good work. This 
course can be abbreviated at the option of teachers as neces- 
sity may require. 

While the advanced course does not aspire to meet the 
requirements of a technical scientific course in the higher insti- 
tutions, yet it is believed that, supplemented by lectures, as 
all text-books should be in the higher institutions, it may meet 
the requirements of the so-called classical courses in many 
colleges. In the high school it provides a way for pupils 
possessed of a special genius and aptitude for scientific studies 
to delve deeper into many subjects than the average pupil is 
wont to do. 

This work is simply a text-book. It lays no claim to be a 
laboratory manual. It is expected that its teachings will be 
supplemented by laboratory work, for laboratory practice has 
come to be considered an essential part of every scientific 
course. Experiments are introduced chiefly for the purpose 
of illustrating principles and laws, but tedious details which 
would tend to distract the attention from, the leading facts 
have been omitted. 

I take this opportunity to acknowledge the debt I owe to 
many able physicists and representative instructors for advice 
and help. My special thanks are due to my friend, Dr. Arthur 
W. Goodspeed, of the University of Pennsylvania, for very 
many valuable suggestions and contributions, and for careful 
reading both of the manuscript and the proof-sheets. Both 
manuscript and proof-sheets have also been critically read by 
Prof. Joseph 0. Thompson, of Amherst College, and my col- 
league, Mr. A. P. Walker, of the English High School, Boston. 
To the former I am under special obligations for much valu- 
able assistance, and to the latter I am largely indebted for 
whatever freedom from rhetorical and typographical errors 
this book may possess. 

Eor valuable criticism of proof-sheets, I have also to record 


my warm personal thanks to Dr. Daniel W. Herring, of the 
University of the City of New York ; Dr. J. S. McKay, of 
the Packer Collegiate Institute, Brooklyn ; Mr. C. F. Adams, 
Detroit High School ; Mr. A. J. Eogers, Milwaukee High 
School ; Prof. J. W. Moore, Lafayette College, Easton, Pa. ; 
Mr. J. T. Coleman, Citadel Academy, Charleston, S. C. ; Mr. 
A. D. G-ray, Penn Charter School, Philadelphia ; and Mr. 
William Orr, Springfield (Mass.) High School. The General 
Electric Company and the Zeigler Electric Company have 
kindly furnished several cuts of electrical machinery and 
other apparatus. 

A key to the solution of problems and exercises contained 
in this book will be furnished by the publishers to those in- 
structors only who use this as their regular text-book. 

A. P. G. 




Fundamental units of measurements. Kinematics. Motion, velocity, 
acceleration. Laws of accelerated motion. Composition and reso- 
lution of velocities. Kinds of motion 1-32 


Chapter I. 

Force and momentum. Measurement of force. Composition and 
resolution of forces. Moments of forces. Center of mass. New- 
ton's Laws of Motion. Curvilinear motion. The pendulum. Work, 
energy, and activity. Machines 33-117 

Chapter II. 
Gravitation 118-123 

Chapter III. 

Properties of matter. Constitution of matter. States of matter. Mo- 
lecular forces. Capillarity. Diffusion of fluids 124-141 

Chapter IV. 

Dynamics of fluids. Transmission of pressure. Atmospheric pres- 
sure. Boyle's Law. Instruments for rarefying air. Siphons and 
pumps. Buoyancy of fluids. Density and Specific density . . . 142-185 


Chapter V. 

Energy of mass-vibration. Sound-waves. Speed of sound-waves. 
Energy of sound-waves. Reflection and refraction of sound-waves. 
Reenforcement and interference of sound-waves. Pitch of musical 
sounds. Composition of sonorous vibrations. Vibration of strings. 
Harmony and discord. Quality of sound. Analysis and synthesis 
of sound-waves. Musical instruments. Vocal organs. The ear . 186-244 



Theory of heat. Sources of heat. Temperature. Thermometry. 
Calorimetry. Effects of heat : Expansion. Kinetic theory of mat- 
ter. Laws of gaseous bodies. Absolute temperature. Fusion. 
Vaporization. Methods of producing cold artificially. Hygrom- 
etry. Diffusion of heat. Thermo-djm amies. Steam-engine . . . 245-314 

PART in. 


Chapter I. 

Radiant energy. Light. Speed of light. Intensity of illumination. 
Apparent size of an object. Reflection of light. Refraction. Prisms 
and lenses. Prismatic analysis of light. Color. Interference and 
diffraction. Double refraction and polarization. Thermal effects 
of radiation. Optical instruments 315-433 

Chapter II. 

Electrostatics. Electrification. Induction. Distribution of elec- 
tricity. Electrical potential. Electrical machines. Electrostatic 
lines of force. Atmospheric electricity 434-461 

Chapter III. 

Electrokinetics. Voltaic batteries. Some defects of batteries. Ef- 
fects produced by the current. Electrical quantities and units. 


Electrostatic units. Rules relating to an electric current. Instru- 
ments for electrical measurements. Resistance of conductors. 
Measurement of resistance. E. M. F. of different cells. Divided 
circuits. Methods of combining voltaic cells. Verification of 
Ohm's Law. Magnets and magnetism. Magnetic lines of force. 
The magnetic circuit. Terrestrial magnetism. Magnetic relations 
of the current. Electro-magnets. Electro-dynamics. Ampere's 
theory of magnetism. Electro-magnetic induction. Dynamo-electric 
machines. Electric motor. The transformer. Storage batteries. 
Electrical transmission of activity. Thermo-electric currents. 
Electro-magnetic theory of light. Electric radiation. Electric 
light. Electrotyping and electroplating. Telegraphy. Telephony. 
The bolometer. Alternating currents. Tesla's investigations . . 462-608 




Accurate knowledge of physical phenomena is obtained 
only by means of precise measurements of physical quan- 

1. Quantity is that attribute of things which makes them 
measurable, i.e. it is that which answers the questions How 
much? How great ? — etc. X quantity i^ therefore a meas- 
urable portion of anything. 

A physical 7)ieasureme7it consists in finding how many times 
a definite quantity, called a unit, is contained in the quantity 
to be measured. Such a unit which has become legalized, 
either by statute or by common usage, is called a standard 

The expression of a physical quantity consists of a state- 
ment of the concrete unit employed, e.g. pound, foot, quart, 
etc., with the number of those units prefixed. The numerical 
part, called the numeric, is obtained by measurement. 

2. The fundamental units, in terms of which all physical 
measurements are made, are those of length, mass, and time} 
These are fundamental in the sense that no one is derivable 

1 '■'■ The whole system of civilized life may be fitly symbolized by a foot rule, a set 
of weights, and a clock." Maxwell. 


from others. The three entities which these units measure 
cannot be absolutely defined ; but, fortunately, they are too 
familiar to require definition. 

The unit of length (or ''space of one dimension") is the 
meter (metric^), or yard (British). The centimeter (see p. 617) 
is the one-hundredth part of a meter. A standard meter is 
defined by law (1795) to be the shortest distance between the 
ends of a platinum rod (made by Borda and called Metre des 
Archives) at 0° C. This rod is in the keeping of the Academy 
of Sciences at Paris. 

The foot is one-third of the British standard yard. The 
yard is defined by act of Parliament (1855) to be the dis- 
tance between the intersections of the transverse lines in 
two gold plugs in a bronze bar deposited at the office of the 
Exchequer in London, the temperature of the bar being 
62° F. (16|° C). 

3. By the mass of a body is meant the quantity of matter 
contained in the body. It is highly important to bear in 
mind that the idea implied in the term onass is quite distinct 
from that of iveight. The weight of a body changes with its 
distance from the earth, while its mass remains the same. 
The weight of a body is the measure of the attraction between 
it and the earth, and is variable because the attraction of the 
earth differs at different places, but its mass is not affected by 
this attraction. 

The unit of mass generally employed in science is the gram 
or the pound. The gram is the one-thousandth part of the 
standard kilogram. This standard is a piece of platinum, 
called the Kilogramme des Archives, carefully preserved by the 

1 The metric system (see page 617) is now generally employed in scientific work. 
The advantage of this system consists largely in the simplicity of the relations which 
exist between the standards of length and mass, and in the use of units each of 
which is some decimal multiple, or sub-multiple, of the others in the same series. 
"The British measurements are infinitely inconvenient and wasteful of brain- 
energy." Tait. — "I look upon our English system as a wickedly brain-destroying 
piece of bondage under which we suffer." Lord Kelvin. 


French Government at Paris. Originally it was intended to 
represent the mass of a cubic decimeter of pure water at the 
temperature of 4° C. A kilogram of any substance is that 
quantity of the substance which, placed on a scale pan, would 
just balance in a vacuum the standard kilogram placed on the 
other pan. 

The English unit of mass is defined by act of Parliament 
(1855) to be a piece of platinum marked "P. S., 1 lb.," 
denominated the Imperial Standard Pound Avoirdupois. It 
is deposited in the office of the Exchequer. 

The process of measuring the mass of a body by balancing 
it with a body or bodies of known mass 'is called weighing. 
The process of weighing, as commonly understood, is essen- 
tially a comparison of masses. A set of masses^ {e.g. a kilo- 
gram down to a milligram) consists of a series of bodies 
having masses corresponding to the denominations given them. 

The process of measuring the mass of a body must not be 
confounded with the process of finding how heavy a body is 
(i.e. how great the attraction between it and the earth is), 
although both processes are, in common usage, called weigh- 
ing. Not only are the things measured in the two cases 
entirely different, but the instruments that may be required 
are quite different. For example, a kilogram mass always 
has the same mass, and may, therefore, when used with the 
scale balance, be relied upon always and everywhere to 
measure accurately an equal mass of any body. But a kilo- 
gram mass when suspended frorn a spring balance may be 
found to be heavier in certain situations than in others, hence 
this instrument does not under all circumstances measure 
mass correctly. On the other hand, the scale balance will 
not detect any change in the heaviness of a kilogram mass ; 
hence this instrument does not always measure heaviness 
correctly. For most practical purposes, however, these in- 

1 Commonly called a set of weights. 


struments may be used intercliangeably, inasmuch as at the 
same place mass is proportional to weight.^ 

The unit of time generally employed in scientific meas- 
urement is the second. The second is „ „1^^ of the mean 

o b -l U U 

solar day. 

4. Derived units ; C. G. S. System. — The system of measure- 
ments in which all the units used in measuring physical quantities 
are derived from the three metric units given above is called the 
centimeter-gram-second system, or, briefly, the C. G. S. system. 
The system is also called an absolute system of units. 

In any absolute system the unit of length is represented symbol- 
ically by [L], that of mass by [M], and that of time by [T]. Any 
derived unit may be represented by certain pov^ers of these symbols, 
or by the product of certain powers of these symbols. Thus the 
unit of area = [L^]. 

5. Dimensional equations. — Any equation showing what powers 
of the fundamental units enter into the expression for the derived 
unit, is called its dimensional equation. The dimensional equation 
for any derived unit is deduced from the physical laws by which 
the unit is defined. / 

6. Volume. — By the volume of a body is meant the quan- 
tity of space it occupies. The derived unit of measurement 
of volume is the cubic centimeter (or cubic foot), and is 
defined as the volume of a cube the length of one side of 
which is one centimeter. 

The dimensional equation for volume is (V) = [L^]. 

7. Matter, body, substance. — Provisionally we may define 
matter'^ as that which occupies space. A body is any limited 

1 This is one of many instances in physics in which one quantity is indirectly 
measured by measuring another proportional to it. Legitimately speaking it is not 
the function of a spring balance to measure mass, nor of the scale balance to 
measure heaviness. The first measures stress (§ 63) ; and the second demonstrates 
equality of moments (§ 50). 

2 Matter is variously defined in scientific text-books according to the fancy of the 
authors. The definitions are, however, only provisional, serving merely the practical 
requirement of distinguishing between what is matter and what is not matter. The 
question "What is matter?" is still a subject of pure speculation, and its discussion 
therefore wholly unsuited to a scientific text-book. The discovery of its ultimate 


portion of matter, e.g. a lake, a tumbler, a desk, etc. Names 
of bodies should be carefully distinguished from names of 
substances which indicate merely the kind of matter of which 
the bodies are composed, such as water, glass, wood, etc. 

8. Phenomena, physical laws. — By observation we learn 
that a piece of iron expands when heated, begins to be 
luminous when heated to a certain temperature, and changes 
to a liquid when heated much more. Here are three distinct 
changes which heat may effect. Changes or events like these, 
and countless others which occur in nature, are called physical 

If, after many trials with iron from various sources and 
under varying conditions, we are able to state in general that 
the application of heat in suitable quantities to iron will be 
followed by these phenomena, such a generalized statement is 
called a physical law. A physical law is an expression of a 
constant relation ivhich has been discovered to exist between 
certain physical quantities. 

Physical laws, unlike statute and moral laws, do not govern 
events, but are generalized statements of the order of events. 
Iron does not expand in obedience to a law that " heat expands." 
We do not explain why a body falls to the earth by stating a 
law that " an unsupported body falls"; indeed, the cause of a 
fall has never been discovered, though every one of us has 
individually discovered the law just quoted. 

nature may be beyond the range of human intelligence. The following attempt to 
answer this question is given (1) to illustrate the significance of the question ; (2) as 
a statement of the theory probably entertained at the present time by our most 
advanced scientists : — " Matter is the rotating parts of an inert perfect fluid which 
fills all space, but which is, when not rotating, absolutely unperceived by our senses." 
Lord Kelvin. 


Sectiojst I. 


9. Motion. — Kinetnatics (from Ktvyjixa, motiofi) treats of 
motions without reference to their causes. Motion is a con- 
tinuous change of position. The position of a particle of 
matter is determined by its direction and distance from another 
particle, or from some point of reference. A particle moves 
relatively to a given point while the straight line con- 
necting it with the point changes either in directio7i or 
length. A particle is at rest relative to a given point while 
a straight line joining them changes neither in direction nor 

While you are opening or shutting the legs of a pair of 
dividers (A, Fig. 1), a straight line a^ V connecting the points 
at the ends of the legs changes in length ; hence there is 
relative motion between these points. If (B, Fig. 1) you 
open the legs a little way, and, fixing the end of one of the 
legs upon a plane surface, trace a circle with the end of the 
other leg around the former as a center, there will be relative 
motion between the two points, since a line joining them, 
ah, ab', etc., changes m direction. 

If (C, Fig. 1) you trace with the points of the open dividers 
two straight parallel lines on a plane surface, the two points 
will be relatively at rest, just as surely as if the dividers were 
lying upon the table, since in both cases a straight line con- 


necting the points a b, a' b', etc., changes neither in length nor 
in direction. 

A point may be at the same instant at rest with reference 
to certain points, and in motion with reference to certain other 
points. For example, while the points of the dividers are 
tracing straight lines on the plane surface (C, Fig. 1), and are 
relatively at rest, they are in motion with reference to every 
point in the plane surface. 

When a particle is spoken of as being in motion or at rest, some 
point is always expressed or understood, relatively to which the 
change or permanence of position is maintained. All motions with 
which we have to do, or which we can measure, are relative. Change 
of position with reference to a fixed point in space would be absolute 
motion. But there is no fixed point which we know. " There are 
no landmarks in space ; one portion of space is exactly like every 
other portion, so that we cannot tell where we are" or in what 
direction, or how fast, we are going. 

In ordinary language the phrase " a body at rest " means that the 
body does not change its position with reference to that on which it 
stands, as, for instance, the surface of the earth or the deck of a 


ship. It can mean nothing else, for both it and all points of the 
earth's surface are in rapid motion with reference to the sun and 
other heavenly bodies, and also with reference to the earth's axis. 

A body moves as a whole with reference to any point only 
when a certain point of that body, called its center of mass, 
or centroid,^ changes its position with reference to the given 
point. Thus the relative motion of two bodies is determined 
by the change of position of their centroids. Likewise the 
path in which a body moves should be understood to mean the 
line described by its centroid. 

10. Velocity/. — No motion is instantaneous. A body con- 
sumes time, longer or shorter, in its transit from one position 
to another. Eate of change of place (or time-rate of displace- 
ment) of a body is called its velocity. Velocity is expressed by 
stating the number of units of distance traversed in a unit of 
time. For scientific purposes it is most frequently expressed 
in centimeters per second ; for practical purposes the units of 
time and distance are chosen at convenience, as the velocity 
of a locomotive in miles ijer hour, of a rifle bullet in feet per 
second, etc. Observe that velocity is dista7ice per unit of time, 
and cannot be correctly expressed in miles, feet, etc., alone. 

Velocity involves the idea of direction, and may change in 
both magnitude and direction. It is sometimes convenient to 
ignore the direction of a body's rate of displacement, in which 
case we use the term speed. Thus it is better to speak of the 
speed of the locomotive and the bullet when only the magnitude 
of the change in position is considered. 

11. Constant and accelerated velocity. — When a body moves 
in a straight line with unchanging rate, i.e. when it traverses 
equal spaces in equal times, its velocity is said to be constant. 
In case, however, of a continuous increase or diminution of 

1 Both the expressions "center of mass" and "center of gravity" are open to 
objections ; hence certain careful writers have suggested as a substitute the term 


velocity, it is said to be accelerated. Finally, if this growth 
or diminution of velocity is uniform, it is said to have constant 

When the velocity increases, as in the case of a falling 
stone, its acceleration ^ is said to be positive, or -f- ; when the 
velocity decreases, as in the case of a stone thrown upward, 
its acceleration is said to be negative, or — . 

Velocity is determined by dividing the distance traversed 

by the time consumed. If a body move s feet in t seconds, its 

s s 

velocity, v, is - feet per second, or ?;==-. In case the velocity 

be accelerated, this result is to be regarded as the average 
velocity for that distance ; and in the case of uniform motion 
the average velocity is the same as the actual velocity at every 
instant. It is evident that the actual velocity of a body whose 
motion changes can be given only at some definite instant or 
point in its journey. It denotes the space which tvould be 
traversed in a unit of time, if at the given instant the velocity 
should become constant. 

In the C. G. S. system the unit of velocity is that rate of dis- 
placement at which a unit of length is traversed in a unit of time, 
and the unit chosen is a centimeter per second. Its dimensional is 
[L/T, or LT-i]. 

The change of velocity of a particle per unit of time is called 
its rate of acceleration, or simply its acceleration, and is 
represented by a. When a particle acquires equal changes 
of velocity in equal units of time its acceleration is said to be 
constant, and its motion uniformly accelerated. The accelera- 
tion of a body falling in a vacuum, and of a body projected 
vertically up in a vacuum is practically constant ; in the 
former case it is about 32.2 feet (or 9.8 m) per second, in the 

1 Acceleration etymologically means an increase of speed, but for convenience 
it has lately come to be applied in scientific treatises to either an increase or a 
decrease of speed. 



latter case it is a negative acceleration of about 32.2 feet 
per second. 

The average acceleration, a, of a particle in traversing a 
certain distance in a given time, and tlie rate of acceleration 
(also rej)resented by tlie symbol a), provided the acceleration 
is constant, is found by dividing the entire change in velocity, 
V, in a certain time by the time, t^ taken in making the change, 


i.e. a = -, whence v = a t. Thus, if the velocity of a rail- 
road train at a certain instant be 25 miles per hour, and 
half an hour hence it be 15 miles per hour, then the entire 
change of velocity, v, is — 10 miles per hour ; hence the 
average acceleration, i.e. the acceleration if it were uniformly 

distributed throuarhout the 30 minutes, is ^,, =( ] oi a 

30 ^ 3^ 

mile per minute. Again, if a stone falling with a constantly 

accelerated velocity acquire in 4 seconds a velocity of 128.8 feet 

128 8 
per second, its acceleration is '^ ' = 32.2 feet per second. 

In the C. G. S. system the unit of acceleration is that acceleration 
in which a unit of speed is gained or lost per unit of time. Its 
dimensional is [LT~2]. 

Section II. 


12. First Lata. — If a particle be moving . at a certain 
instant at a rate V, and its acceleration be + <^? then its 
velocity, v, at any instant is expressed as follows : 

At the initial instant, v = F ; 

At the end of the first unit of time, v=V-\- a X 1 ; 

At the end of two units of time, v ^V-\- a X 2 -, 

At the end of t units of time, v ^V-\- at (A). 

The last, (A), is a general equation expressing the relation 


between the velocity (v) at the end of any given unit of 
time (t), the original velocity (V), and the acceleration (a). 
Erom this formula we derive the following law : 

(1) Change of velocity due to uniform acceleration is equal to 
the product of the acceleration and the units of time. Hence the 
change of velocity is proportional to the rate of acceleration, 
and to the time occupied. 

13. Second Laiu, — If its initial velocity, V, be zero, i.e. if 
the particle start from a state of rest, the equation becomes 
v = at. 

Since the velocity of a particle starting from a state of 
rest increases from zero to a t, the average velocity must be 

— - — =^^at. At this rate in the same time, t, it would 

traverse a distance, S, equal to ^ at X t^=^ at''^ units ; hence 
8=^^ a t^ (B). Prom this we derive the law: (2) The distance 
traversed in a given time by a particle starting from a state of 
rest and having uniformly accelerated velocity., is one half the 
product of the acceleration and the square of the units of time. 
Hence the entire distance traversed is proportional to the 
square of the time, and to the acceleration. 

If a particle, instead of starting from a state of rest, have 
an initial velocity, F, it would move in t units of time without 
acceleration a distance V Xt \ to this distance must be added 
the distance it moves in consequence of acceleration, in order 
to obtain the entire distance traversed in t units, and our 
formula becomes S^^Vt^^at'^ ( 6') . 

14. Verification. — The two laws given above are verified 
approximately and conveniently by the use of the venerable 
Atwood's machine.^ The equal weights A and B (Fig. 2) are 
suspended by a thread passing over the 'wheel C. Inasmuch 
as the weights are equal they counterbalance each other and 

1 This macMne is a contrivance which enables lis to increase the mass to be moved 
without increasing the force which moves it, thus so decreasing the acceleration as 
to render approximate measurements feasible. 



remain at rest. Raise the weight A and 
place it on the platform D as shown in Fig. 3. 
Place on this weight a small additional one, 
E, called a " rider/' the weight of which sets 
the system in motion. Set the pendulum F 
swinging. At each swing it causes a stroke 
of the hammer on the bell G. At the instant 
of the first stroke the pendulum causes the 
platform D to drop so as to allow the weights 
to move. When the weights reach the ring H, 
the rider, not being able to pass through, is 
H caught off by the ring. Eaise and lower the 
ring on the graduated pillar I, and ascertain 
by repeated trials the average distance the 
weights move between the first two strokes 
of the bell, i.e. during one swing of the 
pendulum. Inasmuch as all swings of the 
pendulum are made in equal intervals of time, 
we may take the time of one swing as a unit 
of time. We will also, for convenience, take 
for a itnit of distance the distance the weights 
move during the first unit of time, call this 
unit a space, and represent the unit graphically 
by the line a h (Fig. 4). 

Next ascertain how far the weights move 
from the starting point during two units of 
time, i.e. in the interval of time between 
the first and third strokes of the bell. The 
distance will be found to be four spaces, 
or four times the distance that they moved 
during the first unit of time. This distance 
is represented by the line a c. 

Now ascertain the velocity which the 
weights have at the end of the first unit of 

Fig. 2. 




time. Place the ring H at the 
point (h) which the weights have 
been found by trial to reach at 
the end of the first unit of time. 
Allow the weights to descend as 
before. At the end of the first 
unit of time the rider is caught off. 
At this instant acceleration ceases 
and the motion becomes uniform. 
Ascertain how far the weights move 
with uniform velocity during the 
second unit of time ; this velocity 
is evidently the velocity which the 
weights have at the end of the first 
unit of time. This distance will be 
found to be (approximately ^) two 

Fig. 3. 

1 U.of T. h. 

2 U.of T. 

3 U.of T. 

1 space. 

Represents the velocity at the end of the' 
of time ; also the acceleration during 
unit of time. 

first unit 
the first 

Velocity at the end of the second unit of time. 
Acceleration during the second unit of time. 

Velocity at the end of the third unit of time. 
Acceleration during the third unit of time. 

4 U.ofT. e. 

Fig. 4. 

1 Approximately, since they are retarded by the resistance of the air and the 
friction of the wheel. 


spaces ; hence tlie velocity at the end of the first unit of time 
is two spaces per unit of time. But the velocity at the beginning 
of the first unit of time was zero, hence the acceleration during 
the first unit of time is ttvo spaces per unit of time. 

In like manner determine the velocity at the end of the 
second unit of time. It will be found to be four spaces per 
unit of time. And as the velocity at the end of the first unit 
of time was two spaces per unit of time, the acceleration 
during the second unit of time is two spaces per unit of time. 
Hence the acceleration during the first two units of time is 
uniform, and the change of velocity during the first two units 
of time, as stated in law (l), = a^ = 2 X2 = 4 spaces per unit 
of time. 


1. The velocity of a particle at a certain instant is V ; its acceleration 
is a ; what will be its velocity, -u, in t units of time afterward ? 

2. If the initial velocity of a body be F, its acceleration a, and its 
final velocity v, how long, ^, was it in acquiring its final velocity ? 

3. If a body having an initial velocity V acquire in t seconds a velocity 
V, what was its acceleration ? 

4. If a body move from a state of rest with a uniform acceleration a, 
what space, /S, will it traverse in t units of time ? 

5. If a body move from a state of rest with an acceleration a, in what 
time, t^ will it traverse the space S ? 

6. The velocity of a particle at a certain instant is 20 feet per second ; 
its acceleration is 3 feet per second ; what will be its velocity 10 seconds 
hence ? 

7. Suppose that the acceleration of the particle mentioned above is 
— 2 feet per second, what will be its velocity 5 seconds after the instant 
named ? 

8. a. A body falls from rest ; its velocity increases (if we disregard 
the resistance of the air) 32.2 feet per second. What is its velocity at 
the end of the first second ? 6. What, at the end of the tenth second ? 
c. What, at the end of half a second ? 

9. If the initial velocity of a body be 5 feet per second, its final 
velocity 26 feet per second, and its acceleration 2 feet per second, what 
was the time consumed in acquiring the final velocity ? 


10. A bullet is projected vertically upward with an initial velocity of 
161 feet per second ; what will be its velocity at the end of the third 
second (a = — 32.2 feet per second) ? 

11. How long will the bullet named in the last problem rise ? 

12. What velocity will the bullet have at the end of the sixth second, 
and in what direction will it be moving ? 

13. a. What distance will a body fall from a state of rest in one 
second ? b. In two seconds ? c. In ten seconds ? 

14. A stone thrown vertically downward is given an initial velocity of 
40 feet per second. How far will it descend in ten seconds ? 

15. a. A bullet is projected vertically ujDward with an initial velocity 
of 225.4 feet per second ; how long will it rise ? b. How far will it rise ? 

16. How long will it take a body to fall 1030.4 feet from a state of 

17. a. A body falls during 1^ seconds ; what is its final velocity ? 
b. How far does it fall ? 

18. A body falls 297.6 feet in 4 seconds ; what was its initial velocity ? 

19. What initial velocity must be given a body that it may rise 6 
seconds ? 


Section III. 


15. Gra]}hical representation of ^motion and of velocity. — 
If a person wish to describe to you the motion of a ball struck 
by a bat, he must tell you three things : (1) ivhere it starts, 
(2) in what direction it moves, and (3) how far it goes. These 
three essential elements may be represented graphically by a 

straight line. Thus, suppose balls 

at A and D (Fig. 5) to be struck by 

D ^ E bats, and to move respectively to 

B and E in one second. Then the 
points A and D are their starting-points ; the lines A B and 
D E represent the direction of their motions, and the lengths 
of the lines represent the distances traversed. In reading, the 
direction should be indicated by the order of the letters, as 
AB and DE. The lengths of these lines are not equal to the 
distances traversed by the two balls, but represent these 
distances drawn to some convenient arbitrary scale ; thus on 
a scale of 1 cm = 10 m, these lines represent distances of 32 
and 20 meters respectively. 

The velocity of a moving body is described by giving (1) its 
directioii, and (2) the units of distance per unit of time. Since 
the lines AB and DE represent the distances traversed by 
the two balls during the same unit of time, these lines like- 
wise represent their average velocities during this time, i.e. A B 
represents an average velocity of 32 m^ per second, and D E an 
average velocity of 20 m per second. 

16. Composition of simultaneous velocities. -, — If a particle 
have by any means two or more separate and independent 
motions communicated to it simultaneously, and if the motions 
imparted be themselves constant in velocity and direction, the 
result of their concurrence is a single motion in a straight line 


with a single velocity and direction. This is illustrated some- 
what imperfectly in the following manner. With the handle A 
in the position shown in Fig. 6, push it forward carrying the 
frame B C to the right. This frame carries a pencil D whose 
point presses the paper below, and as the frame advances, the 
line a b is traced upon the paper, graphically representing the 
motion of the pencil. If, when the pencil point is at a and 
the frame is at rest, the string G be pulled, the pencil will 
trace the line ac at right angles to ab. Kow these two 
independent motions may be communicated to the pencil 
simultaneoiisly by fastening the string E to the binding screw F 
and pushing forward the handle A. The pencil point will not 

move in either of the lines ab or ac, but its motion will be 
intermediate between the two, and it will trace the line a d. 
This single motion, which is the result of the concurrence of 
two motions, is called their resultant ; and they, with regard 
to the resultant, are called its comioonents. 

The distance a cl is traversed in exactly the same time that 
the distance ab would be traversed if the pencil had no 
other motion, the handle A being pushed forward with the 
same speed in both cases ; likewise the distance ad is 
traversed in the same time that the distance (xc is accomplished, 
when the string is simply pulled over the pulley G with the 
same speed, and has no other motion. The lines ab, ac, 
and ad represent not only the distances traversed in the 



several directions, but also the magnitudes and directions of 
the respective velocities. For example, if the velocity be 

constant and the pencil reach 
successively at the end of equal 
intervals of time the points 7?^", 
n^', and d (Fig. %a), then am", 
?/2," 7i'\ and n" d represent its 
velocities in the successive inter- 
vals, and a m, m n, and n h repre- 
sent the velocities for the same 
intervals in the direction a h ; and a m', m' n\ and n^ c the 
velocities in the direction ac. 

If points G and d, and d and h are joined by (dotted) lines, 
we have a parallelogram of which the line ad, representing 
the resultant, is a diagonal. Hence to find the resultant of 
two simultaneous velocities when they make an angle with 
each other, the rule is : Construct a parallelogrcun of ivhich 
the adjacent sides represent the two velocities, and the 
diagonal ivhich lies between these adjacent sides rejjresents their 

When more than two com- 
ponents are given, find the re- 
sultant of any two of them, then 
of this resultant and a third, 
and so on until every component 
has been used. For example, let 
the several velocities imparted ^, 
to a particle be represented by 
the lines AB, AC, AD, and 
^^ (Fig. 7). The resultant of 
A B and ^ C is ^ F; the result- 
ant oi AF and AD i^ A G ; 
that oi AG ?iTidi AF i^ AH 
which represents the resultant of the four velocities. 


When two components are at right angles to each other, it 
is evident that we may obtain the magnitude of the resultant 
by finding the square root of the sum of the squares of the 
two components. 

In case a particle has several velocities imparted to it, all 
in the same direction, their resultant is the sum of all. If 
some are opposite others, one of the two directions is con- 
sidered as positive and the opposite direction as negative, 
and these signs being prefixed to the numerical values, their 
algebraic sum is the resultant. 

17. Resolutio7i of a velocity into components. — Any motion 
or velocity may be resolved into two or any given number of 
motions of velocities. Let A B (Fig. 8) repre- 
sent the velocity and direction of motion of a 
particle. Draw a line AC to represent, either 
arbitrarily or according to the conditions of 
the problem, one of the required components. 
Connect B and (7, draw A D parallel with B C, 
and D B with A C, and thus complete a '^^^' ^' 

parallelogram of which ^^ is a diagonal. The two adjacent 
sides A C and AD represent two component velocities of 
the particle ; in other words, a particle having a velocity 
represented by the line AB has at the same time velocities 
represented in magnitude and direction by the lines A C 
and AD. 


1. a. If a ship move east at the rate of 10 miles an hour, and a person 
on deck walk towards the bow at the^rate of 2 miles an hour, what is the 
resultant of these two velocities ? h. With reference to what has he this 
velocity ? 

2. Suppose the person mentioned above walk aft at the rate of 2 miles 
an hour, what will be the resultant of these two velocities ? 6. Prefix 
suitable signs to the numbers given and represent the addition v^^hich 
gives the resultant. 


3. a. A particle moYes simultaneously northward with velocity a, and 
southward with velocity 6 ; what is the resultant of these velocities ? 
6. How do you interpret the resultant if a >► 6 ? c. How, if 6 >► a ? 
d. How, if a =6? 

4. Suppose the person mentioned above walk directly north across the 
deck at the rate of 4 miles an hour, what will be the resultant of these 
two velocities ? 

5. Suppose the person walk northeast at the rate of 4 miles an hour, 
what will be his resultant velocity ? [In drawing the parallelogram of 
velocities, represent the component velocities to some scale, e.g. I of 1 
inch or 1 cm = 1 mile, then having completed the parallelogram and 
having drawn the diagonal which represents the resultant, measure the 
latter and the result will express, on the scale chosen, the resultant 
velocity required.] 

6. Suppose an attempt be made to row a boat at the rate of 6 miles an 
hour directly across a stream flowing at the rate of 10 miles an hour ; 
determine the direction and velocity of the boat. 

7. A vessel sails south-southeast {i.e. 22.5° east of south) at the 
rate of 14 miles an hour ; determine its southerly and its easterly 

8. Represent graphically, to scale, a velocity of 100 feet per second 
and resolve this velocity into two components which shall have an angle 
between them of 45°. 

9. Represent graphically velocities, all in different directions, which a 
particle has at a given instant, as follows : 20 feet, 30 feet, 15 feet, and 
25 feet, per second. Determine its apparent velocity and direction. 

.18. Composition of constant ivith accelerated velocity. — 
Experience teaches that a body, e.g. a stone, projected in a 
horizontal direction moves not in a horizontal path, but in a 
path intermediate between a horizontal and a vertical one, 
showing that its velocity is composed of a horizontal and a 
vertical component. Its horizontal velocity (if the resistance 
of the air be disregarded) is censtant and its vertical velocity 
is uniformly accelerated. Let AB (Fig. 9) represent the verti- 
cal component of the motion during the first second, then B C 
and C D will represent its vertical motion during the second 
and third seconds respectively. Let AB', B'C, and CD' rep- 
resent successive horizontal motions during the same three 



periods. Then it is evident by a combination of these two 
motions that the body will pass from A to B" during the first 
second, from B" to C" during the second second, and from C" 
to D" during the third second. The body traverses a curvi- 
linear path called a parahola, as shown in the figure. In 
practice, the resistance of the air would modify the nature 

of the curve somewhat, so that its real path is a peculiar 
curve known in the science of gunnery as a hallistic curve or 

It should be borne in mind that one of the component 
velocities of a particle moving in a curvilinear path is always 

Problem. — Imagine a body to be projected obliquely upward at an 
angle of 45° ; represent arbitra.rily its vertically downward accelerated 
motion, and its obliquely upward constant motion for three seconds, 
and determine the actual path traversed by the body during this 


Section IV. 


19. Motion of translation and rotation. — In pure motion of 
translation all the points of a body move with the same 
velocity and in the same direction (Fig. 10). Example : the 


Z X_-A 

Fig. 10. Fig. 11. 

Rectilinear motion of translation. Motion of rotation. 

motion of an elevator or a piston in the cylinder of a station- 
ary steam engine. When the points of a body describe arcs 
of circles having its centroid for a common center, the motion 
is one of pure rotation (Fig. 11). Example : the motion of a 
wheel or a top. All possible varieties of motion may be pro- 
duced by the combination of translation and rotation (Figs. 12 
and 13). Examples : the motions of the planets, that of a 
ball thrown from the hand, that of a carriage wheel along 
a road. 

When a body rotates, every particle in the body describes a 
circle around some point or line which is the center or axis of 

The velocity of a point far from the axis is greater than 
that of a point nearer the axis, and, generally, the velocity 
of a point is proportional to its distance from the axis ; 
hence the expression "velocity of a rotating body" is mean^ 
ingles s. 



20. Angular velocity. — We may, however, speak of the angular 
velocity of the rotating body, which is the same for all points in the 
body. Angular velocity is rate of rotation, and is measured by the 
angle turned through by the rotating body in any given unit of time. 

The unit angle in terms of which angular velocity is measured is 
called a radian, and is the angle which is subtended by an arc equal 
in length to the radius. 

It is customary then to express angular velocity of a body in 
radians per second, and this is numerically equal to the speed of a 

Fig. 12. 
Translatory and curvilinear motion. 

Fig. 13. 

Combination of translatory and 

rotary motion. 

point one unit distant from the axis. The Greek letter w (pro- 
nounced o-meg'-a) is chosen to represent angular velocity. 

Now since the linear velocity of any point in a rotating body is 
proportional to its distance, r, from the axis, it must be the product 
of the angular velocity of the body multiplied by the distance of the 
point from the axis, i.e. v = r u. 

It remains to develop a formula for finding the value Of w in terms 
of the time of a complete rotation. It is known that the ratio 
between the circumference of a circle and its diameter is about 3^, 
and is usually represented by the Greek letter tt. Now if the 


body make one rotation in T seconds, the angular velocity is 
equal to the total angle, 2 tt radians, divided by the time, T, 

or co = — radians per second. 


1 . A body makes ten rotations per second ; what is its angular 
velocity ? 

2. What is the actual velocity of a point in this body ten inches 
from the axis of rotation ? 

3. a. What is the angular velocity of the earth's rotation ? 
6. Does its angular velocity vary at different latitudes ? c. Does 
the actual velocity of points on the earth's surface vary at different 
latitudes ? 

4. What kind of motion is that of the earth in its orbit ? 

5. Why is it meaningless to speak of the velocity of rotation of a 

6. a. What motions have the wheels of a carriage drawn straight 
along a level plane ? h. What motion has the carriage ? 

7. Compare the several velocities of the small front wheels of the 
carriage with those of the larger hind wheels. 

8. If a wheel make 200 i^evolutions per minute, what is its 
angular velocity ? 

21. Rectilinear and curvilinear motion. — Besides change in 
velocity or rate of motion, there may be a change in direction 
of motion (see p. 6). When a particle moves in a constant 
direction, i.e. in a straight line, as in the case of a freely fall- 
ing bnllet, its motion is said to be rectilinear. Bnt if its 
motion constantly changes in direction, i.e. at every point, aS' 
is the case of every particle in a rotating wheel except points 
on its axis, its motion is said to be curvilinear. It is 
evident that the direction of a motion in a curvilinear path 
can be given only for some specified point, and, further- 
more, that direction can be represented only by a straight 
line, for a curved line is a line composed of an infinite 
number of directions. Let A (Fig. 14) represent a body 
mounted on a cardboard sector S S^ which is rotated about 

co:mposition of circular motion. 


tlie axis C in the direction indicated by the arrow. The 
body will move in the circular path ADEF. The straight 
line AB will indicate the direction of the motion at every 
point, but it will be seen that this line changes its direction 
constantly. At whatever point the body may be at any 
instant, the line A B, which shows the direction of the motion, 
is always tangent to the curve at that point. 

Fig. 14. 

Fig. 15. 

22. Composition of circular motion. — If a particle move in 
a circular path, e.g. a stone whirled in a sling, its motion 
every instant is the resultant of a tangential motion and a 
centripetal (toward the center) motion. If when it passes 
point A (Fig. 15) its tangential velocity be represented by A B, 
its centripetal velocity may be represented by B C, because at 
the end of the unit of time in which it would reach B if it 
were moving in a straight line, it is found to be not at B, but 
at some other point, C, nearer the center by the distance B C. 

23. Simple Harmonic Motion. — Besides the motions of 
translation and rotation already considered, a third kind of 
motion must be studied somewhat in detail, as it plays so 
important a part in subjects to follow, notably sound and 



Fig. 16. 

If a lead bullet A (Fig. 16) be sus- 
pended by a thread and set swinging 
in a horizontal circular path, the 
motion is practically uniform, and, 
when viewed directly from above or 
below, appears circular. But if the 
motion of the bullet be viewed by 
the eye placed on the same level 
with it, it seems to travel to and fro 
in a straight line, but with varying 
speed. It is seen to move slowly 
near the ends of its path and more 
rapidly in the middle. The motion 
of the bullet as now viewed is virtu- 
ally the projection of uniform cir- 
cular motion on a diameter. 

Thus, let Fig. 17 represent a particle moving with uniform 
speed in the circle ADMN, as in the case of the swinging 
bullet. A line drawn from this particle (at the instant it 
passes the point A in the 
circle) perpendicular to the 
diameter M N, intersects it 
at the point a. While the 
particle moves to B, C, D, 
E, F, and M, this intersect- 
ing point moves to ^, c, d^ 
e,f, and M. Although the 
speed of the moving par- 
ticle is uniform, moving 
over the equal spaces A B, 
B C, CD, etc., in equal 
intervals of time, the speed 
of the intersecting point is 
variable, moving in the same intervals of time respectively 

Fig. 17. 


through the unequal spaces ab, be, cd, etc. The speed of this 
point is greatest as it passes the center of the circle, 
diminishes toward the extremities of the diameter M N, and 
is momentarily zero at the points M and N. Such a motion 
as that, forward and backward along the line MIST, executed 
in equal periods of time, is called simple harmonic motion, or 
S. H. M. 

It is the kind of motion executed by the vibrating prongs 
of a tuning fork, by a stretched string when emitting a 
musical sound, by particles of air when traversed by sound 
waves, and very nearly by a pendulum swinging in a short 
arc. A like motion occurs in the ether when traversed by light 
waves or electrical waves, and in every elastic medium when 
set in a tremor ; hence its intimate relation with many 
branches of physics. 

As the motion repeats itself in regular intervals, it is said 
to be periodic. The time occupied by a particle in executing 
a single complete harmonic motion, i.e. from M to IST and 
back, is called a period. The period is, evidently, the time 
occupied by one complete revolution in the circle of reference, 
as N, D, M, C , ]Sr. When the body appears to move from left 
to right its motion is said to \)Q positive ; and when from right 
to left, negative. The extent of the vibration on either side 
of its middle point, as M or IST, is called the amplitude of 
the S. H. M. The distance of a moving particle from the middle 
point at any instant, as e, is called its displacement, and 
points M and N are points of greatest elongation. The position 
of the particle at any instant is denoted by its phase, which 
is defined as the fraction of a period since the particle last 
passed through in the positive direction. 

24. Composition of Simple Harmonic Motions. — Simple harmonic 

motions in the same or different directions may be compounded 

according to tlie same laws as uniform motions, accelerations, etc. 

We consider first the composition of S. H. M.'s of the same period. 

The resultant of two or more S. H. M.'s of the same period and in 



the same direction is a S. H. M. of the same period as that of the 
components and having an amplitude equal to the sum of the 
amplitudes of the several component motions. This assumes that 
the phases are the same. If, hov^ever, a particle be subjected to 
two S. H. M.'s along the same line, but differing in phase by half a 
period, the resultant will be a motion with an amplitude equal 
to the difference of the amplitudes of the components. That is, 
the particle will remain at rest if the component amplitudes 
be equal. 

If the two motions be in the same phase or in opposite phases 
(i.e. having a difference of phase of one-half of a period) and at 
right angles to each other, the amplitude of the resultant will be 
represented by the diagonal of a rectangle constructed upon those 

lines as adjacent sides which rep- 
resent the amplitudes of the com- 
ponent motions. Thus in Fig. 18, 
let A B and C D represent two com- 
ponent S. H. M. 's at right angles to 
each other, the center of the circle of 
reference being at 0. Then E G or 
r H will represent the resultant mo- 
tion if the phases are alike or oppo- 

If the difference of phase be only 
i period, then the resultant motion 
will be a circle, or an ellipse, accord- 
ing as the component amplitudes are equal or not. If the difference 
of phase be a fraction of a period not as simple as J or i, then the 
resultant motion in general will be an ellipse with its center at 0, 
the position of its major axis depending on the relation of the com- 
ponent amplitudes and on the difference of phase. 

Experiment 1. — The principles given above may be verified 
experimentally by the use of an approximately simple pendulum. 
Partly iill a small glass funnel (Fig. 19) with fine dark writing sand 
and suspend it from a frame above by means of a string (say) 1 m 
long. It can be made to oscillate at will in any direction and the 
sand falling upon a horizontal surface, e.g. a large sheet of white 
paper or pasteboard, leaves a tracing of the motions of the pendulum. 
The position of rest of the bob being at (Fig. 18), give it an impulse 
with the hand in the direction OA sufficient (say) to carry it to 


A. We may, for convenience, divide the complete motion into four 
parts, each of i period, e.g. from to A, A to 0, to B, and B to 
0. A similar impulse may be given to the bob in any other direc- 
tion, for example in the direction C. The motions of 
such a pendulum are approximately S. H. M.'s. The 
directions A and C will be considered as positive, 
and their opposites, B and D, negative. 

We will now examine four cases : (1) While the bob is 
oscillating in the path A B, gently tap it with the hand 
in the direction C at the instant it passes in the positive 
direction ; the bob will now move along the line E, the 
diagonal of the square on A and C. In this case the 
component motions are in the same phase, since both 
impulses are given at and in positive directions. 

(2) While the bob is moving through toward A, let 
the second impulse be in the direction D ; the resultant 
motion is now in the direction H, and the bob oscillates '■ 
along the line F H without change of period. In this ^^^- ^^^ 
case the phases are opposite, differing by i period. 

(3) Let the impulse be in the direction C, but applied when the 
bob is at A instead of at 0. The motion now is in a circle, the 
direction being anti-clockwise (i.e. opposite to the direction in 
which the hands of a clock move). Here the component B A is ^ 
period ahead of the component D C. 

(4) If the impulse be given at A in the direction D, the resultant 
will be a circle still, but traversed in the opposite direction, the 
component B A being now J period behind the component D C. 

Observe that circular motion may be regarded as compounded of 
two S. H. M.'s at right angles to each other, the phases differing by 
J period. 

If the component amplitudes be not equal, the resultant in the 
case of the same or opposite phases will be in a straight line, the 
diagonal of a rectangle, not of a square. With the difference of 
phase of J period, the circle becomes an ellipse with its axes coinci- 
dent with the directions of the component motions. 

25. Composition of S. H. M.''s of different periods. — The motion 
resulting from the composition of two S. H. M.'s of different periods 
is more or less complicated according to the ratio of the periods. 
We have already discussed the case where the ratio is 1:1. The 
next simpler ratio is 1 :2. 



Experiment 2. — Suspend from two tack staples in a horizontal 
bar a loop of string (say) 2 m long. Slip the end of the loop through 
a small clamp which may be adjusted at any height (Fig. 20). 
|] Adjust so that P = i C P. Cause the 
B pendulums OP and CP to oscillate in 
planes at right angles to each other. As 
CP is 4 OP, the period of CP is twice 
that of OP (p. 78). By following the 
directions given in the last experiment, 
the resulting motions may be observed 
and studied. Fig. 21 represents the path 
of P when the phase differences are re- 
spectively 0, i, i and f period. 

Should the ratio be not exactly 1:2, 
the different curves will gradually change 
from one to another. The result of com- 
pounding S. H. M.'s of other period ratios 
may be studied in a similar manner by 
slippiflg the clamp up or down and thus 
changing the relative lengths of the two 
Fig. 20. pendulums. The figures described on the 

paper by the falling sand are exceedingly intricate and interesting. 

Fig. 21. 

26. Composition of S. H. M. with a uniform motion at right 
angles to it ; harmonic curve. — Let the S. H. M. of a particle 
be executed in the line AB (Fig. 22); and let the particle at 
the same time travel with uniform speed from left to right. 
ACB is the circle of reference, which is divided into fourteen 
parts of equal length by the points AC DE, etc. Lines drawn 
through these points at right angles to the line AB determine 
the points on this line (viz., c, d, e, etc.) which define the 
positions of the particle, so far as determined by the S. H. M., 



at equal intervals of J^ of the period. Lines drawn parallel 
to AB divide the space into equal intervals, which represent 
the distances traversed by the particle from left to right 
during each ^l of a S. H. M. period. Now combining the 
motion AM with Ac, MN with cd, NP with de, etc., we get 
points 1, 2, 3, etc., which represent the actual positions of 
the particle after successive intervals of yL of a period. If 
these points be joined by a gently curving line, there results 
a characteristic curve called a Harmonic Curve. 

A M N P 













. & 












































Fig. 22. 

Experiment 3. —Partly fill the funnel pendulum (Fig. 19) with fine 
sand and suspend it from a frame. Set it swinging like a pendulum bob 
and beneath it move uniformly a sheet of paper at right angles to the 
plane in which the pendulum swings. The falling sand will be deposited 
in a curve which is approximately a harmonic curve. 

The movement of the pendulum is approximately S. H. M., 
hence the harmonic curve is the resultant of the S. H. M. of 
the funnel and the uniform motion of the strip of paper. 

27. Wave 7notio7is. — A, B, C, D, etc. (Eig. 23) represent a 
series of particles lying in the same straight line, e.g. a series 




M N 


Fig. 23. 

of particles of water lying in the smooth surface of a body of 
water. Just below, the same particles are represented as 
moving simultaneously, each in a -circular path, in a vertical 


plane. Particle B is just i of a period behind A, C the same 
interval behind B, and so on. A line drawn through the par- 
ticles in their several positions at the same instant is called 
a wave liiie. As long as the particles continue to move in 
their respective circles, so long will a ivave form traverse the 
series of particles. If a person is favorably placed so that 
he can observe a series of water waves passing him, he will 
perceive that floating blocks of wood move in elliptical paths 
(the circular form is not a necessary attribute of these 
motions), never moving more than a certain distance from 
certain points about which they oscillate. The motions of 
the blocks represent rather imperfectly the curvilinear paths 
in which the particles composing a body of water move while 
that body is agitated by waves. To the observer, ridges and 
furrows of water appear to move along the expanse, but 
objects floating upon the surface are not carried along by 
them, which shows that the appearance is a deception, and 
that the body of water is traversed only by wave forms. 
Observe that, whereas in Fig. 22 the harmonic curve results 
in a combination of an harmonic and a uniform translatory 
motion, in Fig. 23 the wave line results from a transmission 
of circular motion to a series of particles in such a manner 
that the motion of any particle shall be a definite part of a 
period behind that of its predecessor. Particles B and J are 
in positions of maximum displacement in the same direction, 
and particles B and F are in positions of maximum displace- 
ment in opposite directions. The distance from B to J, or 
the distance from any particle to the next particle which is 
in the same relative position in its movement, is called a wave 
length; the distance 5i^is a half ivave-length. 



Section I. 


28. Dynamics is the science which treats of the action 
of force. This science will be treated under three heads : 
(1) Molar Dynamics 5 that is, the dynamics of solids and 
fluids, including the study of sound waves ; (2) Molecular 
Dynamics, including heat ; (3) Ether Dynamics ; that is, 
radiation, including light and electricity. 

For present purposes at least, we may regard the term 
Physics^ merely as a generic term which includes all these 
branches. Hence, Physics is the science which treats of the 
dynamics of masses, molecules, and the ether. 

29. Force. — When a body at rest is set in motion, or one 
which is in motion is accelerated (positively or negatively), 
or when a moving body is deflected from a straight course, 
experience teaches us that there is always a cause, and we 
have also learned to apply to this cause the name force. We 
have also learned that when a body is under the influence of 
a force which tends to cause a change of motion in that body, 
another force must act on the same body if a change of motion 
is to be prevented. 

We get our primitive idea of force from the sense of 
muscular exertion which we experience when, by personal 
effort, we put bodies in motion, or stop bodies that are in 
motion. We transfer this conception by analogy to a change 

1 Physics is often defined as the Science of Matter and Energy, since " In the 
physical universe only matter and energy exist independently of our senses and 
reason." (Tait.) 


of motion observed in any body, and attribute this change to 
an interaction between that body and some other body, animate 
or inanimate. This interaction is always a jpull or a push and 
is accordingly called an attractive or a repellent force. It is 
evident that there can be no pull or push except between at 
least two bodies or two parts of the same body, i.e., there is 
no such thing as a one-sided pull. In other words, when 
there is a pull or a push there are at least two bodies pulled 
or pushed, and it is only for the sake of convenience in speech 
that we are permitted to say that one body pulls and the other 
is pulled. 

It is not possible for a person to pull without being himself 
pulled, or to push without being himself pushed. Appearances 
sometimes seem to contradict the above statements. For 
example, a man standing on a wharf pulls a distant boat by 
means of a rope. The boat moves as the result of the pull, 
but, though he is bracing himself against the wharf, he is not 
willing, perhaps, to concede that he is likewise pulled. Let 
him stand in the boat and pull the rope which is attached at 
the other end to the wharf ; both he and the boat move. 
What body, according to appearances, is pulled in this case ? 
What bodies are actually pulled ? 

We are now prepared for a definition of force. Force is an 
interaction between two bodies (or two parts of the same body) 
causing or tending to cause a change in the motion of each, either 
in direction or in magnitude ; or, more simply, force is that 
which tends to modify motion. 

It should be observed that the above conveys no idea of 
what the real nature or essence of force is, for of this we are 
quite ignorant. Indeed we know of the existence of force only 
by its effects; hence an idle force, i.e. a force producing no 
effect, is an absurdity. 

30. Force not a property of matter. — We speak of force as 
exerted by matter, but, strictly speaking, matter does not of 


itself exert force. Matter must be set in motion or have some 
form of energy (see p. 84) conferred upon it before it can 
exert force, so that force is merely a manifestation of energy. 

31. Action and reaction. — Force is always dual, inasmuch 
as it is always oppositely directed upon two bodies. By a 
conventionality of speech we say that one of the two bodies 
acts upon the other, and the latter reacts upon the former. 
Later on it will be shown that the reaction is always equal to 
the action. 

The wings of a bird act upon the air, giving a certain 
portion of it a rearward motion ; the air reacts upon the wings, 
giving the bird a forward motion. The bat strikes the ball, 
imparting to it an acceleration, the ball reacts upon the bat, 
giving it a negative acceleration. 

32, Time required for bodies to gain or surrender velocity . — 
If a sled on which a child is sitting be suddenly put in motion, 
the child is left in the place from which the sled started. If 
the child and sled are both in motion, and the sled be suddenly 
stopped, the child lands some distance ahead. If the sled be 
started slowly, the child partakes of the motion of the sled, 
and is carried along with it ; and if the sled gradually stop, 
the child's motion is gradually checked, and it retains its 
place on the sled. This shows that masses receive motion 
gradually and surrender it gradually. 

Even very small bodies require time to gain or surrender a definite 
velocity. The sand-blast, employed for engraving figures on glass, 
furnishes a fine illustration of this fact. A box of fine quartz-sand 
is placed in an elevated position. A long tube extends vertically 
down from the botton of this box. The plate of glass to be engraved 
is covered with a thin layer of melted wax. The design is sketched 
with a sharp-pointed instrument in the wax when cool, leaving the 
glass exposed only where the lines are traced. The plate is then 
placed beneath the orifice of the tube, and exposed to a shower of 
sand. The velocity of the sand-grains is not at its maximum at the 
start, but is constantly accelerated till thev reach the plate, where^ 


in turn, their velocity is gradually given up. The wax, on account 
of its yielding nature, gradually brings them to rest ; but the glass, 
notwithstanding its hardness, cannot stop them quite at its surface ; 
and, therefore, it suffers a chipping action from the sand. Thus the 
soft wax affords a protection from the action of the falling sand, for 
all parts except those intended to be cut. A still greater force is 
generally given to the sand by steam blown through the tube. For 
this reason the apparatus is called a sand-blast. Hard metals like 
steel are engraved in the same manner. Yet the hand may be held 
in the blast several seconds without injury. 


What is the difference in the effects of catching a base-ball with hands 
held rigidly extended, and with hands allowed to yield somewhat to the 
motion of the ball ? 

33. Momentum. — A small stone dropped upon a cake of 
ice produces little effect ; a large stone dropped upon the ice 
crushes it. An empty car in motion is much more easily 
stopped than a loaded car. Every one knows that the effort 
to stop a moving body depends to some extent upon the mass 
of the bod}^ We have an instinctive dread of the approach 
of large masses. 

Again, we have a similar dread of masses moving with 
great velocities. A ball tossed is a different affair from a ball 
thrown. Thus we are led to the consideration of the mass of 
a body multiplied by its velocity. This product is called 
momentum. A large mass, moving slowly, has great momen- 
tum, but the same mass will have twice the momentum if 
its velocity be doubled ; again, a small mass, moving swiftly, 
has great momentum, but its momentum is increased in pro- 
portion as its mass is increased. A unit of momentum is the 
momentum of a unit mass moving with unit speed. 

If the motion of a mass of 1 k, having a velocity of 1 m 
per second, is taken as a unit of momentum, then a mass 
of 5 k, moving with the same velocity, would have a 

"cvrpuLSE. 37 

momentum of 5 ; and if the latter mass slionld have a velocity 
of 10 in per second, its momentum would be 5 X 10 = 50. 
Hence, the iiumeric of momentum is found by imdtiplying units 
of mass hy units of velocity ; in other words, the product, MV, 
of a mass, 31, by its velocity, V, is its momentum. 

If the mass and the velocity both be unity, the momentum 
will also be unity ; and the unit of momentum, which has 
received no special name, may be defined as the momentum 
of a unit mass moving with unit velocity, and momentum 
may be defined as rate of mass-displacement. 

The dimensional of the momentum of a body is the product of the 
dimensionals of its mass [M] and its velocity [LT-i], i.e. [MLT-i]. 

Since momentum is a quantity which has direction, 
momentum may be compounded and resolved like motions and 

By experiment, we learn that a given force acting for two 
units of time produces twice the velocity that it does in one 
unit of time, and that the velocity which a given force produces 
is proportional to the time it acts ; hence momentum, when 
the force is constant, is equal to the product of the force and 
the time, i.e. Ft\ oi MV= Ft. If the force be not constant, 
then the momentum must be computed from the average force 
acting. The product M V signifies that the mass-motion, or 
momentum, of a body depends on its mass as well as its velocity. 
The product Ft signifies that the momentum imparted to a 
body depends upon the time (t) during which a force acts 
as well as upon the intensity (F) of the force. We infer from 
the above equation that a definite force acting upon any mass 
for a given time will generate in it a speed whose magnitude 
is inversely as the mass. 

34. Impulse. — The product of the time during which a 
force acts by its mean intensity is called the impulse of the 
force. This term is usually restricted to a force acting for a 


short time, as in a blow given to a ball by a bat. There is no 
propriety in asking : " With what force does the bat hit the 
ball ? " The inquiry may be with reference to the average 
pressure which it exerts on the ball, or, more likelj^, with 
reference to the impulse of the force, i.e. the product of 
the mean intensity and the time it acts. It has just been 
shown that Ft^MV, hence we infer that the impulse of a 
force is measured by the momentum produced. 

Questions and Problems. 

1. What agent is the immediate cause of motion ? 

2. What distinction do you make between velocity and momentum ? 

3. Upon what does the momentum given to a ball fired from a gun 
by the expanding gases depend ? 

4. Inasmuch as equal forces are exerted for the same length of time 
by the gases on the ball and the gun, how will the momenta communicated 
to each compare ? 

5. If there be 25 lbs. of matter in the gun and 1 oz. (yL ib.) in the ball, 
and the gun acquire a maximum velocity of 3 feet per second, what, at 
that instant, is the velocity of the ball ? 

6. Can any body be put in motion in no time ? (Demonstrate from 
ioTmnlsiFt= MV.) 

7. Compare the momentum of a car weighing 50 tons, moving 10 feet 
per minute, with that of a lump of ice weighing 5 cwt., at the end of the 
third second of its fall. 

8. With what velocity must a boy weighing 25 K move to have the 
same momentum that a man weighing 80 K has when running at the rate 
of 10 Km per hour ? 

9. Since Ft = M V, to what is change of momentum proportional ? 

10. If the same force act for the same length of time upon bodies hav- 
ing different masses, to what will the velocities produced be proportional ? 

11. Two boats of unequal masses are brought together by pulling on a 
rope. a. Resistance being disregarded, how will their momenta at any 
given instant compare ? 6. How will their velocities at the same instants 
compare ? 

12. If the motion of the moon in its orbit were to cease, these bodies 
would approach each other. The mass of the earth is about 80 times that 
of the moon. What part of the whole distance between them would 
the moon move before collision ? 


35. Force of gravitation^ iveight. — It has been said that 
the best way of defining gravitation is to ''let a stone 
drop." In this phenomenon we discover evidence of the 
action of ^a force between masses ; this force is called gravita- 
tion, or gravity when the action is between the earth and some 
other mass. Inasmuch as by the action of this force the 
earth and stone are brought together^ this force has been 
assumed to be, and is universally spoken of as, an attractive 
force ; but the probability is that the earth and stone do 
not draw each other together, but are brought together 
through the agency of some surrounding medium, and the 
action is quite as likely to be a push as a pull. The term 
weight signiiies the magnitude of the force of gravity which 
exists between any body and the earth. It is usually 
determined by measuring the pressure which gravity causes 
the body weighed to exert upon a supporting body, e.g. on a 
scale pan, or the distortion Avhich it produces in the supporting 
body, e.g. the elongation of the spring in the spring-balance. 
The units generally employed are the 2^ound and kilogram, 
and are called the gravitation units of force. All forces may 
be measured in the same units. To say that a man pulls a 
boat with a force of one hundred pounds is equivalent to 
saying that he pulls with a force that is equal to the force 
which acts between the earth and a body having a mass of 
one hundred pounds. A force of one pound, then, is an 
abbreviated expression for a force equal to the weight (at the 
locality in question) of one pound of matter. 

Sectiox II. 


36. Force tends to produce acceleration. — Thus the force of 
gravity causes bodies to fall with accelerated velocity ; it 
also transforms the otherwise constant velocity of a body 


projected upward into a retarded velocity. A constant force 
acting upon a free body (i.e. a body which encounters no 
resistances) always 'produces a uniformly accelerated motion. 
This is best illustrated by the fall or ascent of a^body in a 
vacuum, the body being meantime acted on only by the 
constant force of gravity. 

37. Absolute measurement of force by direct observation of 
acceleration and mass. — If a force F be applied to a certain 
mass m for a unit of time, a certain momentum is generated 
in the mass. If the same force be applied to a greater mass 
for the same time, it will move with as many times less 
velocity as the mass is times greater, but the product of the 
mass and the velocity, i.e. the momentum, is the same. That 
is, the same force acting for the same length of time on free 
bodies having different masses may be measured by the change 
of momentum generated by it in a unit of time {e.g. a second), 
since this is constant and depends on nothing but the force. 
That is, F^=ma, in which F represents any constant force 
acting on any mass m, a the acceleration, and m a the rate of 
change of momentum. Force is sometimes defined as the time 
rate of change of m.or)ientum. 

38. Absolute units of force. — A unit of force in the absolute 
system is that which, acting for a unit of time, will give to a 
unit of mass a unit of acceleration. The absolute unit of 
force (in the C. G. S. system) is called a dyne, and is that 
force ivhich in one second is capable of giving to a gram-mass 
an acceleration of one centimeter per second ; in other words, it 
is a constant force of the requisite intensity to impart in one 
second to a gram-mass a velocity of one centimeter per second. 

Any constant force which in one second produces in a mass 
of m grams an acceleration of a centimeters per second must 
be equal to m X a dynes {i.e. F=^m. a'). In physics the letter g 
is generally used instead of the letter a to denote the accelera- 
tion due to the force of gravity. By exact measurement the 


acceleration produced by the force of gravity on free bodies 
(i.e. in a vacuum) is found to be, in the latitude of Boston at 
the level of the sea, 980.4 cm per second. Hence the force of 
gravity acting on a mass of one gram must be (substituting, 
in the equation above, W (weight) for F, and g for a) 
W=7ng = l X 980.4 = 980.4 dynes. ^ Consequently it requires 
a force of 980.4 dynes to support (i.e. prevent from falling) 
a mass of one gram ; or the weight of a gram-mass at sea level 
in latitude 42° is 980.4 dynes. A dyne^ is therefore about -^^^ 
of the weight of a gram-mass, or exactly 9 g^.e ^f the weight of 
a gram-mass at Paris. In the gravitation system the weight 
of a gram-mass is a gram-force, hence 1 gram-force = 980.9 
dynes at Paris. Gravitation units in grams-force at Paris are 
readily changed into dynes by multiplying by 980.9 ; at 
Boston, by multiplying by 980.4 ; and generally by multiplying 
by the value of g at any place. 

From the foregoing statements it appears that the weight 
of the gram-mass varies with locality, e.g. Boston or Paris, 
so that the value of a gravitation unit of force, e.g. a gram- 
force, is variable. This want of definiteness constitutes 
a serious objection to the gravitation system when great 
accuracy is required. The value of the dyne is definite, 

39. Exjjression for the mass of a body in terms of its weight. — 


Since W=7ng, m^ — ; that is, mass is measured by its 

weight in poundals ^ or dynes, divided by the acceleration in feet 
or centimeters per second produced by gravity. Although W 

1 The equation W=:'mg expresses the fact that the number of dynes (of gravity) 
acting on a given mass is g times the number of grams-mass in that body. A similar 
statement holds for other systems of units. 

2 A dyne is a very small force. In expressing a force of considerable magnitude 
the megadyne (a million dynes) is commonly used. 

3 A poundal is that force which in one second is capable of giving to a pound-mass 
an acceleration of one foot per second. 


and g vary with latitude and elevation^ tliey vary proportion- 

ally, hence the ratio — {i.e. the mass) does not change, but 

is constant for the same body. 

A body suspended from a spring balance (see Fig. 24) is found to weigh 
at Paris \ K. Required its weight in dynes. Solution : i K = 500 g : 
500 X 980.9 = 490,450 dynes. 

Required to find the force which, acting for 10 sec. , gave to a mass of 

10 g a velocity of 1000 cm per sec. Solution : F=: ^ = 10 X — — - = 1000 


Required to find the mass in which a force of 1500 dynes produces an 

acceleration of 2 cm per sec. Solution: m =— = -^— — 750 g. 
^ a 2 ^ 

Required to find the acceleration which a force of 2000 dynes can give 

a mass of 4 g. Solution -. a — — — — ^— = 500 cm per second. 

m 4 

40. Measurement of force hy countei'halancing . — The house- 
hold instrument called a spring balance is strictly speaking a 
dynamometer, i.e. a force-measurer. It contains a 
spiral spring as seen in A (Fig. 24) carrying an 
index which moves over a scale as shown in B. 
If a unit of mass (e.g. 1 lb. or 1 K) be hung upon 
the spring, it is lengthened by a certain definite 
quantity. If, grasping the ring in one hand and 
the hook in the other, you lengthen the spring by 
a muscular pull as much as it was lengthened by 
the force of gravity acting on the mass, the infer- 
^^' ^' ence is that the muscular force which you exert 
is equal to the force of gravity exerted on the mass, hence the 
spring balance measures all forces in gravitation units. A 
spring balance might, however, be graduated in dynes so as to 
measure force in absolute units. 

The pound and the gram are primarily units of mass. A 
pound-force is a force equal to the weight of one pound of matter. 


41. Two systems of measurement of force. — We have found 
in tlie foregoing discussions that there are two methods of 
measuring force : one specially adapted to measuring balanced 
forces (see p. 45), called the statical or gravitation system ; the 
other specially adapted to measuring unbalanced forces (see 
p. 46), called the kinetic or absolute^ system ; though a force, 
whether balanced or unbalanced, may always be measured by 
either system. The gravitation system is so called because, by 
it, forces are compared with the force of gravity as a standard. 
The two methods of measuring force give rise to tiuo syster}is 
of units called respectively the gravitation and the ahsohite 
systems, either one of which is easily convertible into the 
other, as shown above. ^ 

Questions and Problems. 

1. A constant force acts on an otherwise freely moving body in a 
direction opposite to that in which it is moving ; how is tlie body's motion 
affected thereby ? Give an illustration. 

2. How does a gravitation unit of force differ from an absolute unit ? 

3. Would a spring balance graduated in grams in Paris and sensitive 
to the smallest changes answer for weighing {i. e. determining the exact 
masses of bodies) in Boston ? 

4. What kind of a motion does a constant force produce on a free 
body ? How has this been shown ? 

5. To what is the acceleration produced in equal masses proportional, 
ie. if m is constant, a will vary as what ? 

6. On what condition will equal forces produce equal accelerations ? 

7. Suppose that you fill a box with sand, place it on a toy cart, pull 
the cart by a string with a constant force along a smooth floor for a 
certain number of seconds, and observe the acceleration given the load 
(cart, box, and sand), then remove the sand and replace it with lead shot; 
how could you tell, by pulling the load with the same force as before, 
when it has the same" mass as the former load ? 

1 Measurements of force in the absolute system are attended with serious practical 
difficulties in the way of observation of the acceleration produced, yet the absolute 
units are almost indispensable in very many scientific calculations, especially in 
electricity and magnetism. 


8. a. Has the same mass equal weights in Paris and Boston ? b. How 
sensitive must a spring balance be to discover any difference ? 

9. Show that a spring balance is, strictly speaking, a force-measurer, 
and not a mass-measurer. 

10. a. When we speak of a force of one pound, what do we mean ? 
6. When we speak of a force of one dyne, what do we mean ? c. When 
we speak of a mass of one pound, what do we mean ? 

11. a. If one mass is four times another, how many times as much 
force is necessary to produce the same acceleration in the former as in 
the latter ? b. How many times greater is the force of gravity acting on 
a mass of one hundred pounds than on a mass of one pound ? c. If a 
hundred-pound iron ball and a one-pound iron ball be let drop from the 
same hight at the same instant, which ought to reach the ground 

12. A body weighing 4 g is moving with an acceleration of 12 cm per 
second ; what is the force acting ? 

. 13. A body acted on by a force of 100 dynes receives an acceleration 
of 20 cm per second ; what is its mass ? 

14. A body of m.ass 30 g is moved by a constant force of 50 dynes ; 
what is its acceleration ? 

15. What force acting on unit mass for unit time.will cause it to move 
with unit velocity ? 

16. What acceleration will a force of 20 dynes produce on a mass 
of lOg? 

. 17. What velocity will a force of 20 dynes acting on 1 K impart to it 
in 5 minutes ? 

18. a. What is the weight in dynes of a mass of 1 K in Boston ? 
b. How many more dynes does it weigh in Paris ? 

19. A constant force of 20 dynes acts on a mass of 5 g and gives it a 
velocity of 500 cm per second ; how many seconds does it act ? 

20. How is the value in dynes of a gram weight at any locality deter- 
mined ? 

21. How much is the momentum of a gram-mass changed by gravity 
in one second when falling freely in Boston ? 

22. Why will a bullet fired at an open door pass through it without 
moving it perceptibly, while a push of the hand, of much less intensity, 
moves it an appreciable distance ? 

23. Explain why the weight of a body is not a perfect measure of its 

24. What is the relation of the static unit of force to the kinetic 


Section III. 


42. Grajpliical representation of force. — A force is defined 
when its magnitude, direction, and 'point of application, are 
given. Hence we may represent forces graphically by straight 
lines whose lengths bear to one another the same relation as 
the numerics of the forces, while the directions of these lines 
indicate the directions of the forces, and the points from 
which the lines are drawn indicate the points of application. 
Thus, on a scale of 1 cm =: 1 k the 
line A B (Fig. 25) represents a force 
of 3.2 k acting toward the right with ^ ^ ^ 

its point of application at A ; and ^^^' ^' 

the line D E represents a force of 2 k acting parallel to the 
first with its point of application at D. 

43. Composition of forces acting hi the same line ; equilibiHum 
of forces; balanced and unbalanced forces. 

Experiment. — Insert two stout screw-eyes in opposite extremities of 
a block of wood. Attach a spring balance to each eye. Let two persons 
pull on the spring balances at the same time, and with equal force, 
as shown by the indexes, but in opposite directions, — The block does 
not move. One force just neutralizes the other, and the result, so far as 
any movement of the block is concerned, is the same as if no force acted 
on it. 

When one force opposes in any degree another force, each 
is spoken of as a resistance to the other. Let / represent the 
number of pounds of any given force, and let a force acting 
in any given direction be called positive, and indicated by the 
plus (+) sign, and a force when acting in an opposite direc- 
tion to the force which we have denominated positive, be called 
negative, and indicated by the minus ( — ) sign. Then if two 
forces +/ and — / acting on a body at the same point or along 


the same line are equal, they are said to be balanced, and the 
result is that no change of motion is produced. 

Viewed algebraically, +/ — /^ ; or, correctly interpreted, 
-\-f—f=z= (is equivalent to) 0, i.e. no force. In all such 
cases there is said to be an equilibrium of forces, and the body 
is said to be in a state of equilihriwn. 

A force that produces equilibrium with one or more forces 
is called an equilibrant. 

If one of the forces be greater than the other, the excess is 
spoken of as an unbalanced force, and its direction is indicated 
by one or the other sign, as the case may be. Thus, if 
a force of -f- 8 pounds act on a body toward the east, and 
a force of — 10 pounds act on the same body along the same 
line, then the unbalanced force is — 2 pounds ; i.e. the 
result is the same as if a single force of 2 pounds acted on 
the body toward the west. Such an equivalent force is called 
a resultant. A resultant force is a single force that may be 
substituted for ttuo or more forces and produce the same result 
that the simultaneous action of the several forces would produce. 

The resultant of any number of forces acting m the same 
straight line is equal to the algebraic sum of the forces. An 
equilibrant of several forces is equal in magnitude to their 
resultant, but opposite i7i direction. The process of combining 
several forces so as to find their resultant is called composition 
of forces. The forces combined are called the components. 
The converse operation, of finding component forces which 
shall have the same effect as a given force, is called resolution 
of forces. 

44. An unbalanced force alivays pjroduces acceleration. — 
A body acted on by an unbalanced force cannot be at rest. 
That branch of dynamics which treats of the relation of force 
to the motion which it produces is called kinetics, and that 
branch which treats of equilibrium of forces is called statics. 

Equilibrium is often maintained by the reaction of a sur- 


face with which the body acted on is in contact. A simple 
illustration is that of a body supported on a horizontal sur- 
face, as of a table. Here the reaction caused by the com- 
pression of the material of which the table is composed is 
equal to the weight of the body. 

45. Pressujx, tension. — A balanced force does not produce 
acceleration, but causes either a pressure or a tension. A force 
exerts pressure when it tends to compress or shorten in the 
direction of its action the body on which it acts. Examples : 
Pressure exerted on the springs of a carriage, on air when it is 
compressed in an air gun, etc. A force causes tension when it 
tends to lengthen in the direction of its action a body on which 
it acts. A body thus subjected to a force tending to elongate 
it is said to be in a state of tension., and the stress to which it 
is subjected is called its tension, and the strength to resist 
being pulled apart which it possesses is called its tensile 

Questions and Exercises. 

1. Explain the use of a line to represent force ? 

2. a. When a force of 100 lbs. is represented by a line 5 inches long, 
what is the scale ? 6. "What force will a line J in. long represent on the 
same scale ? 

3. a. Represent on a scale of i inch = 1 lb. the resultant of forces of 
5 lbs. and 7 lbs. acting in the same direction. (Always place arrow 
heads in lines representing forces to indicate the direction of the forces.) 
b. Show, by points A, B, and C placed in the line, the components of this 
resultant, c. Represent the same two forces acting in opposite directions 
upon the same point A. d. How will you represent the resultant of 
these two opposing forces ? 

4. Three men, A, B, and C, pull on a rope in the same direction with 
forces respectively of 50 lbs. , 60 lbs. , and 70 lbs. A is nearest the end 
of the rope, B next, and C next. a. What is the tension of the rope 
between A and B ? 6. What, between B and C ? c. A man, D, just 
beyond C pulls with a force of 75 lbs. .in the opposite direction. With 
what force must a man, E, pull, that there may be equilibrium ? 
d. When there is equilibrium, what is the tension of the rope between 


C and D ? e. How great must be the tensile strength of the rope 
between C and D ? /. Write the equation showing the algebraic ad- 
dition of the forces in case of equilibrium. 

Section IV. 


46. Composition of ijarallel forces acting in the same 
direction and in the same ^jZa?ze. 

Experiment. — AB (Fig. 26) 
represents a rod lying on a table 
with three strings loosely looped 
around it so that they may be 
slid along the rod. Dynamom- 
eters are attached to the free 
ends of the strings. The strings 
are all stretched in parallel direc- 
tions in a plane parallel with the 
top of the table. (Great care must 

Fig. 26. 

be taken in the manipulation to keep the three strings exactly parallel.) 
The dynamometers register the tensions in the several strings, i.e. the 
forces applied through them to the rod. 

Observe (I) when there is equilibrium the dynamometer 
E registers as much as those of F and G added together. 
(This would be true if more than two forces were applied in 
the same direction as A F and B Gr.) But the force applied at 
C is the equilibrant of the other forces and this is equal to 
their resultant acting in the direction CD. (II) The point 
of application of the resultant (or equilibrant) is between the 
points of application of the components. (HI) This point is 
nearer the greater force. (IV) The distance of this point 
from the smaller force is as many times greater than its 
distance from the larger force as the larger force is times the 
smaller force. For example, if A F be 14 lbs. and B G 6 lbs., 


i.e. 14 : 6 — 7 : 3, then distances C A and B will be as 3 : 7. 
In other words the component forces are said to vary inversely 
as,^ or to be inversely proportional to^ their distances from their 
resultant. These observations are summarized as follows : 
The resultant of tiuo parallel forces in the same direction is 
equal to their sum, and the distances of their points of apjplica- 
tionfrom the point of application of the resultant vary iyiversely 
as the intensities of the components. 

Corollary : The condition of eq_uilibriuin is that the algebraic 
sum of the forces (positive and negative) must he zero. 

When more than two forces act on a body in the same 
plane and in the same direction, the resultant of any two of 
them (and its point of application) is found, then the resultant 
of this resultant and a third force, and so on until all have 
been used. 

47. Composition of two unequal parallel forces acting in 
opposite directions. — Let F and F^ (Fig. 27) be parallel forces 
acting in opposite directions on C B, of which F is the greater. 
The force F may be resolved into two 
forces : one, represented by BBi, equal 
and opposite to Fi ; the other equal to 
F — Fi, and represented by the line C D. 
But the forces F^ and B Bi are in equi- 
librium, leaving an unbalanced force at 
C equal to F — Fj. This, then, is the 
resultant E of the forces F and Fi, i.e. 
R=:F— Fi. But BBi:CD=AC:AB, ^ ^ 

and by composition B Bi + C D : B Bi= ^"^^^-^t- 

AC + AB:ACorF:Fi = CB:CA. Hence the resultant of two 
unequal parallel forces acting in opposite directions is equal to 
their difference, and acts outside of both in the direction of the 

1 The pupil should acquire immediate familiarity with these expressions of fre- 
quent occurrence in physics, and should practice in this connection writing inverse 
proportions. Thus for the quantities here given, 14 : 6 — J : i, i.e. the forces are 
proportional to the reciprocals of their respective distances from the resultant. 


greater component, and the distances of its point of appli- 
cation from the points of application of the two forces are 
inversely proportional to their intensities. 

48. Dynamical couple. — If F and Fi (Fig. 27) be equal, the 
magnitude of the resultant, being equal to the difference of the 
components, is zero, i.e. they have no resultant. Two equal 
forces applied to the same body in parallel and opposite directions 
not in the same line constitute what is called a ^^ couple.'^ 
The effect of a couple is to produce rotation, but no motion 
of translation. The value of a couple will be determined 
later on. 

49. Moment of a force. — The value of a force to produce 
rotation around a given axis is called its moment with refer- 

A 3ft. c 3ft. B ^^^^ to that axis. The 

ZS axis is, of course, al- 

Ibs. 20 lbs. V , • 1 , 

ways a line at right 

D ^ angles to the plane 

^^^•28- of rotation. Point C 

(Fig. 28) may represent the extremity of the axis about which 

A B is supposed to rotate. The perpendicular distance (C A 

or C B) from the axis of rotation to the line of direction in 

which a force acts (A D or B E) is called the arm or leverage 

of the force. 

The Tnoment of a force is measured by the product of the 
intensity of the force into the 
arm. For example, the moment / ^"\ '' " "^ 

of the force AD (Fig. 2^) is A/-----_\c/y^^^\,^^^ 
expressed numerically by the / ^^ ^^Q 

number (30 X 2 =) 60, and the / 
moment of B E is (20 X 3 =) 60. > 
BydefinitionthelineAC(Fig.29) ^'''- ^^• 

is the arm of force P, and B C of the force Q. 

50. Equilibrium of moments. — The moment of a force is 
said to be positive when it tends to produce right-hand rota- 



tion, i.e. in the direction in which the hands of a clock move, 
and negative when its tendency is in the reverse direction. 
If two forces act at different points of a body which is free to 
rotate about a fixed pointy they will produce equilihriuifn when 
the algebraic sum of their moments is zero. Thus the moment 
of the force applied at A (Fig. 28) is — (30 X 2) = — 60. 
The moment of the force applied at B in an opposite direction 
is accordingly + (20 X 3) = + 60. Their algebraic sum is 
zero, consequently there is equilibrium between the moments, 
and no tendency to rotation. 

When more than two forces act in this manner, there will 
be equilibrium if the sum of all the positive moments be 
equal to the sum of all 
the negative moments. 
Thus, the sum of the 
positive moments act- 
ing about point D (Fig. 
30) is (/) 45 + (e) 25 
+ (a) 30 = 100 ; the 
sum of the negative moments acting about the same point is 
(c) 30 + {d) 40 + {b) 30 = 100 ; the two sums being equal, the 
moments are in equilibrium. 

51. Moment of a coupjle. — The moment of a couple, or its 
Pi value in producing rotation, is the sum of 
the moments of its two components around 
the axis of rotation. Let F and Fi consti- 
tute a couple whose arm is AB (Fig. 31). 
To find the rotating value of the couple, let 
P be the axis of rotation, then the moments 
of F and Fi relatively to P are FxAP, 
and Fi X B P. The total resultant moment 
of the two forces is (F X AP) + (Fi XB P), or (since F^Fi) 




5 D 

xo ''■' 











Fig. 30. 

Fig. 31. 


Questions and Problems. 

1. Two parallel forces of 8 lbs. and 12 lbs. act in the same direction 
respectively at points A and B, 12 inches apart. Find the magnitude 
and position of their resultant. 

2. ■ The smaller of two parallel forces having the same direction is 5 
inches from the resultant ; what is the distance of the resultant from the 
other force ? 

3. Two men carry a weight of 100 lbs. suspended from a pole 15 feet 
long ; each man is 18 inches from his end of the pole. Where must the 
weight be attached in order that one man may bear f of it ? 

4. Take from the last problem the number of pounds supported by 
each man and the respective distances of each from the weight, and make 
an inverse proportion which shows the relation that must exist between 
these quantities. 

5. How can a force of 4 lbs. be made to produce equilibrium with a 
force of 12 lbs. ? 

6. Draw a line 2 inches long. Kepresent on a scale of i inch = 1 lb. 
a force of 8 lbs. applied at a point A :^ of 1 inch from one end of the line 
and at right angles to it. Take for the axis of rotation a point B f inch 
from the same end of the line. From point C i inch from the other end 
of the line draw a line which will represent a force that will produce 
equilibrium with the first force, and thereby prevent rotation. 

7. Repeat the work of the last problem except that the force applied 
at A shall act obliquely on the line. 

8. Can a single force produce equilibrium with a couple ? 

9. a. A plank weighing 40 lbs. is placed across a log so as to be 
balanced. A boy weighing 60 lbs. sits on one end of the plank. Where 
shall another boy weighing 90 lbs. sit that he may balance the first? 
b. What pressure will be exerted upon the log ? 

10. Two horses harnessed abreast are ploughing. How can you 
arrange that one horse shall pull only two-thirds as much as the 
other ? 

11. The maximum muscular force which a certain man can exert is 
200 lbs. With what leverages can he raise a stone weighing a ton ? 

12. How can pressure be multiplied indefinitely ? 



Section V. 


52. Center of mass defined. — Let Fig. 32 represent any 
body of matter ; for instance, a stone. Every particle of the 
body is acted upon by the force of gravitation. The forces of 
gravitation of all the particles form a set 
of parallel forces acting vertically down- 
ward, the resultant of which equals their 
sum (§ 43), and has the same direction as its 
components. The resultant passes through 
a definite point in whatever position the 
body may be, and this point is called its 
center of mass, or centroid. The center 
of mass (cm.) of a body is, therefore, the 
point of application of the resultant of 
all these forces ; and for practical purposes the whole Tnass of 
the body may be supposed to be concentrated at this point. ^ By 
the place or location of a body mathematicians mean that 
point where its center of mass is situated. 

It is evident that in whatever position a body be placed, 
the resultant of the lueights of all its particles passes through its 
centroid. Hence, to support a body (i.e. to prevent its falling), 
the supporting force, or equilibrant, — or the resultant of several 
supporting forces, — mMst act in a line through the centroid of 
the body and vertically upward. A vertical line is any straight 
line passing through the centroid of the earth. Up and down 
are directions in this line from and toward the earth's centroid. 

Let G in the figure represent the cm. of the stone. For 
practical purposes, then, we may consider that the force of 

^. ^ The expression center of mass does not necessarily signify that point occupying 

■ * a central position among the particles of a body, but a point where, for convenience 

in dynamical problems, we may consider all the mass (or inertia) to be concentrated. 


gravitation acts only at this point, and in the direction GF. 
If the stone fall freely, this point cannot deviate from a 
vertical path, however mnch other points of the body may 
rotate about this point during its fall. Inasmuch, then, as the 
cm. of a falling body always describes a definite path, a line 
GF that represents this path, or the path in which a body 
supported tends to move, is called the line of direction. It 
may be defined as a straight line in which lie the centroid 
of the body and the centroid of the earth. 

To support any body, then, it is only necessary to provide a 
support for its centroid. The supporting force must be applied 
someivhere in the line of direction. The difficulty of poising a 
book, or any other object, on the end of a finger, consists in 
keeping the support under its centroid, i.e. in the line of direction. 

Fig. 33 represents a toy called a " witch," consisting of a cylinder 
of pith terminating in a hemisphere of lead. The toy will not lie in 
a horizontal position, as shown in the 
figure, because the support is not 
applied immediately under its cm. at G-\ 
G ; but when placed horizontally it 
immediately assumes a vertical po- 
sition. It appears to the observer to 

rise ; but, regarded in a technical sense, it really falls, because its 
cm., where all the mass is supposed to be concentrated, takes a 
lower position. 

Whether a body having no other support than that applied at 
its base ivill stand or fall depends up)on whether or not its line 
of direction falls luithin its base.' The .base of a body is not 
necessarily limited to that part of the under surface of a body 
that touches its support. For example, place a string around 
the four legs of a table close to the floor : the rectangular 
figure bounded by the string is the base of the table. (What is 
the base of a man when standing on one foot ? on two feet ?) 

The centroid of any symmetrical body of homogeneous 
material (i.e. of uniform density) coincides with its geo- 



metrical center. Examples : the middle point of a material 
straight line ; that point on a straight line joining the vertex 
to the middle of the base of a triangle situated at a distance 
from the vertex eqnal to two-thirds the length of the line ; the 
geometrical center of any polygon, a sphere, a circular cylinder. 

53. Hoiv to find the center of mass of a body. — Imagine 
a string to be attached to a potato by means of a tack, as in 
Fig. 34, and to be suspended from 
the hand. When the potato is at 
rest, there is an equilibrium of forces, 
and the cm. must be Somewhere in 
the line of direction an ; hence, if a 
knitting-needle is thrust vertically 
through the potato from a, so as to 
represent a continuation of the verti- 
cal line oa, the cm. must lie some- 
where in the path an made by the 
needle. Suspend the potato from 
some other point, as h, and a needle thrust vertically through 
the potato from h will also pass through the cm. Since the 
cm. lies in both the lines an and hs, it must be at c, their 
point of intersection. It will be found that, from whatever 
point the potato is supported, the point c will always be verti- 
cally under the point of support. On the same principle the 
cm. of any body is found. But the cm. of a body may not be 
coincident with any particle of the body; for example, the 
cm. of a ring, a hollow sphere, etc. 

54. Three states of equilibrium. — That a body acted on 
solely by the force of gravitation may be in equilibrium, it is 
necessary and sufficient that a vertical line through, its centroid 
shall pass through the point or surface by which it is sup- 
ported. The weight of a body is a force tending downward ; 
hence, a body tends to assume a position such that its cm. will 
be as low as possible. 

Fig. 34. 


Experiment. — Try to support a ring on the end of a stick, as at b 
(Fig. 35). If you can keep the support exactly under the cm. of the ring, 
there will be an equilibrium of forces, and the ring will remain at rest. 
But if it is slightly disturbed, the equilibrium will be destroyed, and the 
ring will fall. Support it at a ; in this position its cm. is as low as pos- 
sible, and any disturbance will raise its cm. ; but, in consequence of the 
tendency of the cm. to get as low as possible, it will quickly fall back into 
its original position. 

A body is said to be in stable equilibrium if its position is 
such, that any motion except of translation would raise its cm., 
since in that event it would tend to return 
to its original position. On the other hand, 
a body is said to be in U7istahle equilibrium 
when a disturbance would lower its cm., 
since it would not tend to return to its 
original position. 

A body is said to be in neutral or indiffer- 
ent equilibrium when it rests equally well 
in any position in which it may be placed. 
A sphere of uniform density, resting on a horizontal plane, 
is in neutral equilibrium, because its cm. is neither raised nor 
lowered by a change of base. Likewise, when the support is 
applied at the cm., as when a wheel is supported by an axle, 
the body is in neutral equilibrium. 

It is evident that if the c.rn. he below the support, as in the 
last experiment with the ring, the equilibrium must be stable; 
but a body may be in stable equilibrium, though its 
cm. be above the point of support. (When is this 
possible ?) 

It is difficult to balance a lead-pencil on the end of 
a finger ; but by attaching two knives to it, as in Eig. 
36, the cm. may be brought below the support, and 
it may then be rocked to and fro without falling. ^^^- 36. 

55. Stability of Bodies. — The ease or difficulty with which 
bodies supported at their bases are overturned varies with the 



hight to which their cm. must be raised to overturn them. 
The letter c (Fig. 37) marks the position of the cm. of each of 
the four bodies A, B, C, and D. If any one of these bodies 
be overturned, its cm. must have passed through the arc ci, 
and have been raised through the hight ai. By comparing A 
with B, and supposing them to be of equal weight, we learn 
that of two bodies of equal weight and hight of, the cm. of 
that body ivhich has the larger base viust be raised higher, and 
that body is, therefore, overturjied with greater difficulty. A 

comparison of A and C, supposing them to be of equal weight, 
shows that ivhen two bodies have equal bases and lueights, the 
body having its cm. higher is more easily overturned. D and C 
have equal masses, bases, and hights, but D is made heavy at 
the bottom, and this lowers its cm., and gives it greater stability. 

Questions and Exercises. 

1. Where is the centroicl of a box ? 

2. Why is a pyramid a very stable structure ? 

3. What is the object of ballast in a vessel ? 

4. State several ways of giving stability to an inkstand. 

5. a. In what position would you place a cone on a horizontal plane, 
that it may be in stable equilibrium ? b. That it may be in neutral 
equilibrium ? c. That it may be in unstable equilibrium ? 

6. In loading a wagon, where should the heavy luggage be placed ? 
Why ? 



7. Why are bipeds slower in learning to walk than quadrupeds ? 

8. Why is mercury placed in the bulb of a hydrometer ? 

9. How will a man rising in a 
boat affect its stability ? 

10. Which is more liable to be 
overturned, a load of hay or a load 
of stone of equal weight ? 

11. Draw a triangle and find its 
center of mass. 

12. What attitude does a man 
assume when carrying a heavy load 
on his back ? Why ? 

13. Explain the difference in the 
behavior of a ball and of a cube, 
when placed on a plane slightly in- 

14. What position do bodies floating in air or in water take ? 

15. a. Explain how the toy horse (Fig. 38) stands upon the platform 
without falling off. 6. Explain how the toy may rock upon its support 
without falling off. 

Fig. 38. 

Section VI. 


bQ>. When the handle A (Fig. 6) is pushed forward, there 
is applied to the pencil a force which may be represented in 
magnitude and direction by the line a h ; at the same time 
the pencil is pulled vertically up by a force which may 
properly be represented by the line a c. The pencil, however, 
moves in the line a d, which is a diagonal of a parallelogram 
constructed on the lines ah and etc. It is evident that a single 
force might be applied to the pencil with the same effect that 
the two forces produce. Obviously, if a single force were to 
move the pencil in the line a d, it must have the direction of 
this line. It remains to ascertain whether the diagonal line 
ad represents the magnitude of the resultant. Evidently if 



this diagonal does represent the resultant, then the same 
diagonal with the direction reversed will represent the 
equilibrant of these forces. We put the matter to an experi- 
mental test with other apparatus : 

Experiment. — Insert pegs in any three holes of the circle in the top 
of the circular table, Fig. 39. Join these by threads attached to a spring 

Fig. 39. 

balance as shown in the figure. Stretch the balances so as to indicate 
any desired pull in each of the threads. Place under the threads a sheet 
of white paper. Locate on the paper the common point of application A 
of the three forces. Draw lines AB, AC, and AD, to represent the 
directions in which the forces act. Since the point A does not move, it 
is evident that the three forces are in equilibrium and that any one of the 
three forces is the equilibrant of the other two. Select any one for an 
equilibrant {e.g. AD) and extend it in' an opposite direction from A, 



representing (on some suitable scale) a force A E equal to and opposite to 
the force A D as indicated by the dynamometer D, On the same scale 
lay off distances A B and A C representing the magnitudes of the forces 
acting in the directions of these lines. The line AE is by definition 
(§ 43) the resultant of A B and A C. Connect E with C and B. The 
figure, if the work be done with care, will be found to be a parallelogram. 
The diagonal EA represents the magnitude of the equilibrant of the 
forces AB and AC, and the same line with the direction reversed (i.e. 
A E) represents the resultant. 

57. Parallelogrmn of forces. — If two forces applied at a 
point he represented in magnitude and direction hy the adjacent 
sides of a paradlelograrn, their resultant luill be represented in 
magnitude and direction hy the diagonal which passes through 
that point. 

This proposition is applicable whether the forces act on a 
particle or on a rigid body provided they lie in the same plane. 

Thus, let two forces applied at points A and B of a stone 
(Fig. 40) act in the directions A C and B D respectively. 
The direction of the resultant must pass through E, the point 
where the lines of direction of the given forces produced back- 


wards intersect. If, now, the lines E C and E D be laid off to 
represent the relative intensities of the forces, the diagonal 
E F of the parallelogram constructed thereon will represent 
their resultant, and its point of application may be G or any 
other point in the line G H. 

58. Composition of more than tiuo forces in the same plane. — 
When more than tivo components are given, find the resultant 
of any two of them, then of this resultant and a third, and so on 
till every component has been used. Thus, in Fig. 41, A C is 
the resultant of A B and A D, and __— — ^B 

A F is the resultant of A C and A E, A^f^c:::;^'^ ^^ ' 

i.e. of the three forces A B, AD, / 1^\~~^^~^"^^^~-~~^ ', 
andAE. (Invent several problems / \ ^n.-— — ""^^^^ 
similar to this, in which three, four / d \. 

or more forces are to be combined, ~~^~~~-~~.^^ ^s,^^ / 

and work out the results.) ~ p 

Generally speaking, a motion ^i^- *i- 

m.ay he the result of any number of forces. When we see a 
body in motion, we cannot determine by its behavior how 
many forces have concurred to produce its motion. 

59. Triangle of forces. — Since in Fig. 39 BE = AC, the three 
forces which are represented in the parallelogram by the lines A B 
AC, and A E, are also represented by AB, BE, and AE, three 
sides of a triangle ABE. 

Hence, if two forces are represented by two sides of a triangle, the 
third side will represent their resultant. 

60. Polygon of forces. — If any 
number of forces applied at a point 
are represented by all the sides but one ^■= 
of a polygon, the remaining side will 
represent their resultant. Thus the 
forces AB, AD, and AE (Fig. 41), 
are represented respectively by the 
sides A' B', B'C (=AD), and C'F' 
( = A E) of a polygon A' B' C F' ^^^- *^- 

(Fig. 42), that is completed by the side A'F' (= AF), which rep- 
resents the resultant of the three forces. 



61. Parallelopiped of forces. — If three forces not in the same 

^ plane are applied at a point, they 

"":^:^^^^ will form three edges of a par al- 

---,---^^ iQlgplpQ^^ ^^(^ ffifj^^ diagonal of 

'\^ '\ this solid which is concurrent with 

'\ \ these edges will represent the re- 

D^\W>^ ^'-,:^^ sultant of these forces. It will be 

A B readily seen that the resultant of 

^^^•^^- the forces A B, AD, and AC 

(Fig. 43), is represented by the diagonal AE. 

62. Resolution of forces. — Assume that a ball has an 
acceleration in a certain direction A C (Fig. 44), and that one 
of the forces that produces this acceleration is represented in 
intensity and direction by the line A B ; what must be the 
intensity and direction of the other force ? Since AC is the 
resultant of two forces acting at an angle to each other it is 
the diagonal of a parallelo- 
gram of which AB is one 
of the sides. From C, draw 
CD parallel and equal to 

B A, and complete the paral- ^ * '^ 

lelogram by connecting the 

points B and C, and A and D. Then, according to the 
principle of composition of forces, A D represents the inten- 
sity and direction of the force which, combined with the force 
A B, would move the ball from A to C. The component A B 
being given, no other single force than A D will satisfy the 

Had the question been. What forces can produce the motion 
AC? an infinite number of answers might be given. In a 
like manner, if the question were. What numbers added 
together will produce 50 ? the answer might be 20 + 30, 
40 + 10, 20 -p 20 -|- 10, and so on, ad infinitum ; but if the 
question were, What number added to 30 will produce 50 ? 
only one answer could be given. 



It is often necessary to resolve a force in order to ascertain the 
effective force in a certain direction. Thus when boat sails are 
exposed obliquely to the wind, the pressure effectual in moving the 
boat is only a component of the whole force of the wind. The line 
af (Fig. 45) represents the force of the wind acting on the sail c cZ at 
the point a. Resolving this force we obtain the components 2 

(normal to the sail) and! (a useless component called a tail wind). 
The boat does not move in the direction of the pressure on its sail, 
because it is more easily moved lengthwise than breadthwise. Hence 
the normal pressure must be resolved into two components, one 4 
along the direction of least resistance, i.e. the direction of easy 
motion, the other 3 at right angles to it. The latter component 
does tend to cause a slow broad-side motion called leeway, but this 
may be partly counteracted by a deep keel or a center-board so that 
the boat will sail approximately along the line a b. 


1. Draw upon paper pairs of lines making about the s^me angles with 
each other as A B and A C in the four diagrams, Fig. 46, and having 

B B 

about the same directions ; assign arbitrarily numerical values to each 
component, drawing to scale, and find the direction and the numerical 
value of the resultant of each pair of components. 


2. a. Find the intensity of the resultant of two forces acting at an 
angle of 45°. 6. Find the intensity of their resultant when they act at an 
angle of 150°. (The pupil will require either a pair of dividers or a pro- 
tractor. He will do well to learn to use both in measuring angles.) 

3. a. A heavy rock rests upon a smooth plane ; two men, A and B, 
pull the rock by means of ropes attached to it, A with a force of 100 lbs. , 
B with a force of 150 lbs. If A pull toward the north and B toward the 
south, what will be the resultant ? 6. If A pull toward the east and B 
toward the south, what will be the resultant ? c. In the last case, if the 
easterly acceleration at a certain instant is 10 feet per minute, what is 
the southerly acceleration at the same instant ? d. In what direction 
should they pull the rock to give it the maximum acceleration ? e. If A 
pull it 25° S. of E., in what direction and with what force may B pull it 
that the resultant may be directly east ? /. Give a different answer to 
the last question. 

4. On a scale of 1 cm = 1 K, represent a force of 5 K acting north- 
ward on a point A. 

5. On a scale of 1 cm = 1 K, represent forces of 4 K, 6 K, and 8 K, 
acting simultaneously on point A in directions respectively as follows : 
N., N. E., and S. E. Find their equilibrant. 

6. A ship is sailing N. N. E. at the rate of 12 knots per hour. Find 
its northerly and easterly velocities. 

7. Find, both by construction (of parallelogram) and by calculation, 
the intensity of two equal forces acting at right angles to each other, that 
will support a weight of 15 pounds. 

8. A sailor climbs a mast at a uniform rate of 5 feet a minute while 
the vessel moves forward at the rate of 15 feet a minute ; what is his 
actual velocity ? 

9. On a scale of I of one inch = 10 lbs. , represent a force of 80 lbs. 
Eesolve this force into two forces one of which shall act at an angle of 
30° with the given force. Determine the numerical intensities of each of 
the components. 

10. Show by construction that a north-east wind is made up of a 

north and an east wind, each — p of the actual velocity of the wind. 


11. If two lines AB, C A represent two forces acting on point A, the 
one toward and the other from it, show how to find the resultant. 

12. Find the resultant of two equal forces of P lbs., the angle between 
them being 120°. 

13. Two rafters, making an angle of 60°, support a chandelier weighing 
90 lbs. ; what is the pressure along each rafter ? Ans. 51.96 pounds. 


Section VII. 
DISCUSSION OF Newton's three laws of motion. 

63. Laius of motion. — The science of dynamics rests on 
certain fundamental principles termed the Laws of Motion, 
first clearly stated by Newton in the ^' Principia " two cen- 
turies ago, and verified by universal experience. The laws as 
given in this text-book are as originally enunciated by New- 
ton, with very slight verbal modifications in conformity to 
modern terminology. 

First Lata : A body at rest remains at rest, and a body in 
motion continues to move with constant speed in a straight line, 
unless acted upon by some external unbalanced force. 

This law may be paraphrased as follows : A body under the 
action of no force, or of balanced forces, is either at rest or in 
uniform motion ; if it be at rest it will remain at rest, and if 
it be in motion, its motion will be in none other than a straight 
line, and its velocity will never change. 

Motion unobstructed is perpetual. "Is perpetual motion 
possible ? " has been often asked. The answer is simple, — 
yes, more than possible, often necessary, if no force interfere to 
prevent. (Example : the motions of the planets.) On our 
earth we have no instances, for resistances such as friction, 
resistance of the air, etc., are continually opposed to all 
movements of terrestrial bodies. On this account we find 
that force is required to perpetuate the motion of all bodies 
with which we deal, and we fall readily into the fallacy that 
force is necessary to Tnaintain motion, which the First Law 
distinctly contradicts. 

The clause " Unless acted upon by an external force " virtu- 
ally states that "All matter is inert," i.e. that bodies of mat- 
ter are utterly incapable of putting themselves in motion or 
stopping themselves ; the inability is called inertia. Inertia 


may be defined as that property of matter in virtue of which 
external force is required to produce a change in momeyitum. 
It is the sole unalterable property of matter. 

The terms mass and inertia are often used interchangeably to 
denote a quantity proportional to the unbalanced force required 
to produce a given change in the velocity of a body in a given 
time. It is known that all bodies unobstructed by the air fall 
with the same velocity irrespective of their masses or inertia. But 
to produce equal acceleration in equal times requires forces propor- 
tional to the masses; it follov^s, then, that at the same locality 
weight and mass {or inertia) are proportional. Hence we compare 
masses by comparing their weights. 

The somewhat vague yet common expressions "to overcome 
inertia" and "to destroy inertia" mean to produce a certain 
change of mass-motion {i.e. momentum), and may signify either an 
increase or a decrease of the same. 

Second Laiv : Change of momentum is in the direction in 
which the unbalanced force acts, and is proportional to its inten- 
sity and to the time during luhich it acts. 

It will be seen that tliis law (except as regards direction) 
is contained in the formula MV=Ft (p. 37) which has 
already been developed. This formula virtually asserts that 
where there is no force there is no change of momentum 
(i.e. if F = 0, Ft = 0). Hence the First Law of Motion is a 
deduction from the Second. 

This law declares, by implication, (1) that an unbalanced 
force in a given time alivays produces exactly the same change 
of momentum regardless of the mass of the body ; that an 
unbalanced force never fails to produce a change of momentum, 
hence any force, however small, can move any body of however 
great mass. For example, a child can move a body having a 
mass equal to that of the earth, provided only that the motion 
of this body is not hindered by a third body. Moreover, the 
quantity of momentum that the child can generate in this 
immense body in a given time is precisely the same as that 


which he would generate by the exertion of the same force for 
the same length of time on a body having a mass of (say) 10 
pounds. Momentum is the product of mass into velocity; 
so, of course, as the mass is large, the velocity acquired in a 
given time will be correspondingly small. The instant the 
child begins to act, the immense body begins to move. Its 
velocity, infinitesimally small at the beginning, would increase 
at an almost infinitesimally slow rate, so that it might be 
years before its motion would become perceptible. 

It is easy to see how persons may get the impression that 
very large masses are immovable except by very great forces. 
The erroneous idea is acquired that bodies of matter are capa- 
ble of resisting the tendency of forces to cause motion, and 
that the greater the mass, the greater the resistance (" quality 
of not yielding to force," Webster). The fact is, that 7io body 
of whatever mass can resist motion ; in other words, " a body 
free to move cannot remain at rest under the slightest unbalanced 
forceP But as tiine is always required to generate change of 
momentum, there arises thence a deceptive appearance of 
resistance or holding back. 

This law declares by implication, (2) that a force acting 
on a body in motion -produces just the same effect as if it were 
acting on the same body at rest, for no reference is made in the 
\2iW to the state of the body acted upon. 

Experiment. — Draw back the rod d (Fig. 47) towards the left, and 
place the detent-pin c in one of the slots. Place one of the brass balls on 
the projecting rod, and in contact with the end of the instrument, as at A. 
Place the other ball in the short tube B, Raise the apparatus to as great 
an elevation as practicable, and place it in a perfectly horizontal position. 
Eelease the detent, and the rod, propelled by the elastic force of the 
spring within, will strike the ball B, projecting it in a horizontal direc- 
tion. At the same instant that B leaves the tube and is free to fall, 
the ball A is released from the rod, and begins to fall. The sounds 
made on striking the floor reach the ears of the observer at the same 
instant ; this shows that both balls reach the floor in sensibly the same 



time, and that the horizontal motion whicli one of the balls has does 
not affect the time of its fall, i.e. does not modify the effect of the force 
of gravity. 

The law implies, (3) that if two or more forces act on a 
body, each ^produces its own change of monientuTn in its own 
direction independently of the others. It declares, what we have 
previously learned, that the operation of compounding forces 

Flo. 47. 

is just the same as that of compounding motions which the 
several forces tend to produce in the same time, hence the 
apparatus, Fig. 6, illustrates either the composition of motions 
or the composition of the forces by which the motions are 

Third Laiu: To every action there is an equal and opposite 

Previous to the announcement of the Laws of Motion our 
studies have been such as to prepare us both to understand 
and accept them. We have learned that there are always 
two bodies or two parts of the same body oppositely affected 
by every force. When the double aspect of a force, i.e. its 


mutual action between two portions of matter, is considered, 
it is customary to speak of the force as a stress. Illustrations 
of stress are tension in a stretched rubber band and pressure 
exerted between two bodies in contact when compressed. All 
force is of the nature of a stress and the Third Law of Motion 
virtually declares that evei^y action hetiueen tivo bodies is a 
stress. When the effect of the action upon only one of the 
two bodies is under consideration, the action is commonly 
spoken of as a force. 

It remains to show that action and reaction are equal. That 
they are equal is deducible from the First Law, for if they 
were unequal, then, when there is an action between two 
parts of the same body, there would not be equilibrium. That 
is, there would be an unbalanced force, which would cause 
the body to move with accelerated velocity — a thing which 
is explicitly contradicted by the First Law. 

If action and reaction were not equal there might be a 
possibility that a person might raise himself by pulling on 
the soles of his feet or the hair of his head; that a vessel 
might be propelled in a calm by blowing against its sail with 
a powerful bellows (operated by steam) located on the deck 
of the same vessel ; that a person sitting in a buggy might 
give himself a ride by pressing his feet against the dasher ; 
that a person might advance, i.e. move his center of mass, 
without the earth beneath him ; that a bird might fly without 
the external air to act on. 

In case the two bodies are free from the action of resisting 
forces the law implies that the momenta generated by the 
action and reaction are equal. 

The application of this law is not always obvious. Thus, 
an apple falls to the ground in consequence of an action 
between the apple and the earth. The motion of the earth 
toward the apple is imperceptible. But this is because 
the mass of the earth is enormously greater than that of 


the apple, and its velocity, for an equal momentum, is 
proportionately less. 


1. a. Why does not a given force, acting the same length of time, give 
a loaded car as great a velocity as an empty car ? b. After equal forces 
have acted for the same length of time upon both cars, and have given 
them unequal velocities, which will be the more difficult to stop ? 

2. a. The planets move unceasingly ; is this evidence that there are 
forces pushing or pulling them along ? b. None of their motions are in 
straight lines ; are they acted upon by external forces ? 

3. A certain body is in motion ; suppose that all hindrances to motion 
and all external forces be withdrawn from it, how long will it move ? 
Why? In what direction? Why? With what kind of motion, i.e. 
accelerated, retarded, or uniform ? Why ? 

4. Explain how rotating lawn-sprinklers are kept in motion. 

5. When you leap from the earth, which receives the greater momen- 
tum, your body or the earth ? 

6. When you kick a door-rock, why does snow or mud on your shoes 
fly off? 

7. If a man in a boat move it by pulling oh a rope at one end, the 
other end being fastened to a post, how is the boat put in motion ? 
Would it move either faster or slower if the other end were fastened to 
another boat free to move, the man exerting the same force ? 

8. An ounce bullet leaves a gun of mass 8 pounds with a speed of 800 
feet per second. What is the maximum speed of the gun's recoil ? 

9. Suspend two balls of soft putty of equal mass, A and B (Fig. 48). 
Draw A to one side, and let it fall so as to strike B. Both balls will then 
move on together ; with what momentum compared with A's momentum 
when it strikes B ? 

10. What will be the momentum of each ball after A strikes B, com- 
pared with A's momentum when it strikes B ? 

11. How will their velocity compare with A's velocity when it strikes B? 

12. Raise A and B equal distances in opposite directions, and let them 
fall so as to collide. Both balls will instantly come to rest after collision. 
Show that this result is consistent with the third law of motion. 

13. Substitute for the inelastic putty balls, ivory billiard balls, which 
are highly elastic. Let A strike B. Then B goes on with A's original 
velocity, while A is brought to rest. Show that this result is consistent 
with the third law of motion. 



14. Suspend four ivory balls, C, D, E, and F. Let C strike D. D 
receives all of C's momentum, instantly communicates it to E, and E to 
F. F, liaving nothing to which to communicate the momentum, moves 
with C's original velocity. Trace the actions and reactions throughout. 

15. What would happen if the four balls were inelastic ? 

IG. A sliell at rest bursts into two parts, the smaller being one-third of 
the whole ; what is the ratio of the initial velocities of the parts ? 

17. a. Can any body, animate or inanimate, by any action confined to 




A B 

A' B' C' 
Fig. 48. 


C D E F 

itself, i.e. between component parts of itself, put itself in motion or stop 
itself ? 6. How can a body put itself in motion ? 

18. A child sits upon a sled. The sled is suddenly started and the 
child is left sitting on the ice. a. Is this due to the inertia of the child ? 
h. Is it due to a resistance which the child's body offers to a force tending 
to put it in motion, or to the inadequacy of the force transmitted to it 
through the sled to give its mass in the same time an equal velocity with 
the sled ? 

19. Why do not heavy bodies fall faster in a vacuum than light 
ones ? 

20. Take equal masses of wood and lead ; which weighs more ? 

21. A stone falls from the top of a railway carriage which is moving at 
the rate of one-half of a mile a minute. Disregarding the resistance of 
the air, find what horizontal distance and what vertical distance the stone 
will have passed through in one-tenth of a second. Ans. 4.4 ft. ; .16 ft. 


Section VIII. 


64. How Gui'vilinear motion is produced. — Motion is curvi- 
linear when its direction changes at every point. But according 
to the first law of motion, every moving body proceeds in a 
straight line unless compelled to depart from it by some 
external force. Hence curvilinear motion can be produced 
only by an external force acting continuously upon the body 
at an angle to the straight path in which the body tends to 
move, so as constantly to change its direction. In case the 
body moves in a circle, this force acts at right angles to the 
path of the body or towards the center of motion ; hence this 
deflecting force has received the name of central force. 

Thus, suppose a ball at A (Fig. 49), suspended by a string 
from a point d, to be struck by a bat, 
in a manner that would cause it to 
move in the direction Ao. At the 
same time it is restrained from taking 
that path by the tension of the string, 
which operates like a force drawing 
it toward d. It therefore takes, in 
obedience to the two forces, an inter- 
mediate course. At c its motion is 
in the direction C7i, in which path it 
would move but for the string, in accordance with the first 
law of motion. Here, again, it is compelled to take an inter- 
mediate path. Thus, at every point, the tendency of the 
moving body is to preserve the direction it has at that point, 
and consequently to move in a straight line. The only reason 
it does not so move is that it is at every point forced from its 
natural path by the pull of the string. But if, when the ball 


reaches the point i, the string be cut, the ball, having no 
force operating to change its motion, continues in the direction 
in which it is moving at that point, i.e. in the direction ih, 
which is tangent to its former circular path. 

65. Magnitude of central force for bodies moving in circular 

Experiment 1. — Cause a ball to revolve around your hand by means of 
a string attached to it and held in the hand. Observe closely every phase 
of the operation. First you make a movement as if to project the ball 
in a straight line. Immediately you begin to pull on the string to prevent 
its going in a straight line. Under the continuous influence of these two 
forces in a short time the ball acquires great speed. You may now cease 
to exert any projecting force, and simply keep the hand still ; but as the 
ball has acquired a motion, and all motion tends to be in a straight line, 
you are still obliged to exert a pulling force to deflect it from its path. 
Observe that, as the velocity of the ball is retarded by the resistance of 
the air, the pulling or deflecting force which you are obliged to employ 
rapidly diminishes. 

To satisfy yourself that the ball tends to move in a straight line, let go 
the string or cut it, and the ball immediately moves off in a straight line, 
or simply perseveres in the direction it had at the instant the string was 
cut. Observe that the ball appears while rotating to be pulling your hand ; 
but you know that all the force concerned originates in yourself, and that 
this apparent pull on the part of the ball is only the effect of the reaction 
of the force which you exert on the ball. This reaction is erroneously 
called "centrifugal force. "i 

Every revolving body affords an example of central force 
and centrifugal tendency. Hence we say that every revolving 
body tends to fly away from the center (not radially, however, 
but tangentially), and a central force is required to keep the 
body in its circular path. 

When you swing the ball about your hand you discover 
that the force of the pull increases with the velocity, and 
more rapidly than the velocity. Careful observations have 

* There is no centrifugal force. The only force exerted is the central force, which 
is of such a magnitude as to change the direction of the momentum just fast enough 
to keep the body moving in a circle. 



determined that for bodies revolving in circular orbits the 
central force varies as the mass of the body, as the square of its 
velocity, and inversely as its distance from the center. 

Let a point move uniformly in the circular path P Q (Fig. 49a), 
traversing the distance P in time t ; 
then OP = v^l). If P be very near 
O, the deflection T P from a straight 
line due to the central force is ap- 
proximately equal to ON. If a be 
the acceleration towards the center 
due to this force, ON = ia^2(2). 
But by geometry 0P2z=0N-0D. 
Comparing (1) and (2), we get v-t^ 


= |a^2.2r, ora = — . Since E == 
m a, we get F = m — . 

The farther a point is from the axis of motion of a rigid 
body, the farther it has to move during a rotation ; con- 
sequently the greater its velocity. Hence, bodies situated at 
the earth's equator have the greatest velocity, due to the 
earth's rotation, and consequently the greatest tendency to fly 
off from its surface. The effect of this is to neutralize, in 
some measure, the force of gravity. It is calculated that a 
body weighs about 2^9 less at the equator than at either pole, 
in consequence of the greater centrifugal tendency at the 
former place. But 289 is the square of 17; hence, if the 
earth's velocity were increased seventeenfold, objects at the 
equator would weigh nothing, i.e. the centrifugal tendency 
would be equal to their weight. 

The attraction between the sun and the earth causes these 
bodies to move in curvilinear paths, performing what are 
called annual revolutions. Were it not for this mutual 
attraction (and the attraction of the other celestial bodies), 
the motion of both these bodies would be eternally in straight 



Fig. 50. 

lines, but in consequence of their mutual attraction both 
rotate about a point C (Fig. 50), which is the center of mass 
of the two bodies considered as one body (as if connected 
by a rigid rod).^ If 
both bodies had equal 
masses, the center of 
gravity and center of 
motion would be half- 
way between the two 
bodies ; but as the mass of the earth is less than that of the 
sun, so its velocity and distance traversed are proportionally 
greater. In reality the center of motion C is within the sun 
near the edge toward the earth. 

Experiment 2. — Apply the frame T (Fig. 51) to any rotating appa- 
ratus as E (Fig. 52) so that it may be rotated about its axis d. The rod c 
passes tlirough the balls a and 6 loosely so that the latter are free to 
slide along the rod. The two balls are connected by a string so that 
they are compelled to rotate as one body or one system of bodies. 

Fig. 51. 

The mass of a is twice that of h. Eotate the system, and show that there 
is equilibrium in the system only when the center of h is twice as far from 
the axis of rotation as the center of a. How does this verify the above 
law ? While there is rotation, is there tension in the string connecting 
the balls ? What is the cause of an action between the balls ? Ball a 
pulls ball h ; what is the effect of this pull on 6 ? What is the effect of 
the reaction on a ? Is there a similar action between the sun and earth in 

1 Strictly speaking, the earth does not revolve around the sun any more than the 
sun around the earth ; but both rotate about their commoh centroid. 



their annual revolutions ? By what name is the action known ? If the 
sun or the earth were instantaneously annihilated, state what would 
happen to the other body if it were left entirely free, i.e. if its motion 
were not affected by other bodies in the universe ? 

Fig. 52. 

Experiment 3. — Arrange some kind of rotating apparatus, e.g. E 
(Fig. 52). Suspend a skein of thread a (Fig. 53) by a string, and cause 

it to rotate ; it assumes the shape of 
the oblate spheroid a\ Mount a glass 
globe G (Fig. 52) about one-tenth 
full of colored water, and rotate. 

C,— «^w . . ». ^ The liquid gradually leaves the bot- 

I '''^^s'^nS^ II /{l» n» ^^°^' ^^^®^' ^^^ forms an equatorial 
d ^Pfc n B 11 wlill////Jr ring within the glass. This illustrates 

the probable method by which the 
earth, on the supposition that it was 
once in a fluid state, assumed its 
present spheroidal state. (Explain.) 
Pass a string through the longest diameter of an onion c, and cause 
it to rotate ; the onion gradually changes its position so as to rotate on 
its shortest axis. 

It can be demonstrated mathematically, as well as experi- 
mentally, that a freely rotating body is in stable equilibrium 

Fig. 53. 


only when rotating about its shortest diameter; hence the 
tendency of a rotating body to take this position. 


1. a. What is the cause of the stretching force exerted on the rubber 
cord when you swing a return ball about your hand ? 6. Suppose that 
you double the velocity of the ball ; how many times shall you increase 
this stretching force ? 

2. In what way can the tension in the string (Fig. 51) be so much 
increased as to break it ? 

3. Why do wheels and grindstones, when rapidly rotating, tend to 
break, and the pieces to fly off ? 

4. On what does the magnitude of the pull between a rotating body 
and its center of motion depend ? 

5. Oj. Explain the danger of a carriage being overturned in turning a 
corner. 6. How many fold is the tendency to overturn increased by 
doubling the velocity of the carriage ? 

6. Account for the curvilinear orbits of the planets. 

7. How are their motions in their orbits and around their axes main- 
tained ? 

8. In what way should the rails be laid so as to neutralize the centrif- 
ugal tendency of a railroad train going around a curve ? 

9. State and explain the posture of a bicycle rider in turning a curve. 

10. In what way is the weight of terrestrial bodies nullified in some 
degree by the earth's motion ? 

11. A circus rider going around a ring inclines inward so that the line 
of direction of his body falls without his base. How is he supported ? 

Section IX. 


Experiment 1. — From a bracket suspend by strings leaden balls, as in 
Fig. 54. Draw B and C to one side, and to different hights, so that B 
may swing through a short arc, and let both drop at the same instant. 
C moves much faster than B, and completes a longer journey at each 
swing, but both complete their swing, or vibration, in the same time. 

Hence, (1) the time occupied by the vibration of a penduliiin 
is independent of the length of the arc. Of only very small arcs 



may this law be regarded as practically true. The pendulum 
requires a somewhat longer time for a long arc of vibration 
than for a short one, but the difference becomes perceptible 
only when the difference between the arcs is great, and then 
only after many vibrations. 

Experiment 2. — Set all the balls swinging ; only B and C swing 
together ; the shorter the pendulum, the 
I faster it swings. Make B 1™ long, and 
F i™ long. Watch in hand, count the 
J) vibrations made by B, It completes just 
G 60 vibrations in a minute ; in other words, 
it " beats seconds." A pendulum, there- 
fore, to beat seconds must be 1"^ long 
(more accurately in the latitude of Boston 
at sea-level, .9935™, or 39.117 in.). Count 
the vibrations of E ; it makes 120 vibra- 
tions in the same time that B makes 60 
vibrations. Make G one-ninth the length 
of B ; the former makes three vibrations 
while the latter makes one, consequently 
the time of vibration of the former is one- 
third that of the latter. 




•'a B C 

Fig. 54. 

Hence, (2) the time of one vibra- 
tion of a pendulum varies as the 
square root of its length. 
The length Z of a simple pendulum to swing in a time t, or 
the time of swing for a length I, can be found from the 
formulae : 

.9935 X f\ whence t = ^ 


or Z = 39.117 X tK whence t 



for I inches. 

The isochronism of the pendulum is utilized in the measure- 
ment of time, i.e. in subdividing the solar day into hours, 
minutes, and seconds. The office of the pendulum in 


clocks is to regulate the rate of motion of the works. The 
balance-wheel replaces the pendulum in watches and some 

66. Determination of the accelerating force of gravitation at 
any locality. — The time of vibration is less at a place where 
the force of gravitation is greater because the accelerating 
force for the same mass is greater and hence the pendulum 
will move faster. 

Hence it is apparent that by determining the time of 
vibration of a pendulum ^ of the same length at different 
distances from the center of mass of the earth (e.g. at the 
top and bottom of a mountain, or at sea-level at different 
latitudes), the relative value of g at these places, i.e. the 
acceleration produced by gravitation, may be ascertained. 
We have already learned that the acceleration at the same 
locality is the same for all bodies regardless of their mass. 

By experiments too difficult for ordinary school work, it has 
been ascertained that (3) the time of vibration of a pendulurn 
varies inversely as the square root of the force of gravitation 
(upon which the value of g depends). 

To sum up the above three laws of the pendulum, we have 
the formula^ 



whence g=^ — ^> 
9 ^ 

in which I = length of pendulum ; t = time of one vibration 
in seconds. 

1 The following measurement of g was made with great care occupying months 
by Mendenhall, at Tokio, Japan, in the year 1880. The latitude of this place is 
N. 35° 41'; value of g at sea-level 9.7984™; length of seconds-pendulum 994.59'°™. 
On the summit of a neighboring mountain 12,441 feet above the level of the sea, he 
found the time of vibration of the same pendulum to be 1.000336 seconds. From this 
he computed the value of ^ = 9.7886". He also calculated the attraction of the 
mountain to be .00021 the attraction of the earth, and that if the mountain were 
annihilated, at that altitude g would be equal to 9.7865 m. 

2 The student may find the development of this formula in Chapter VII of 
Maxwell's " Matter and Motion." 


At the poles of the earth the length of a seconds-pendulum 
is 99.62«™ and ^ = 983.2«'^ per second. At the equator, 
I = 99.10^"^ ; g = 978.1^™ per second (Kohlrausch). 

67. Center of oscillation. 

Experiment 3. — Connect six balls, at intervals of 15cm, y^y passing a 
wire through them, after the manner of pendulum A (Fig. 54). This 
forms a compound pendulum composed of six simple pendulums. Set 
A and B vibrating ; A vibrates faster than B, although their lengths are 
the same. Why is this ? If A v^ere actuated only by the ball/, it v^^ould 
vibrate in unison w^ith B. If the ball a were free, it would move much 
faster than / ; but, as they are constrained to move together, the tendency 
of a is to quicken the motion of /, and the tendency of / is to check the 
motion of a. But e is quickened less than /, and d less than e ; on the 
other hand, h is checked by / less than a, and c less than 6. It is apparent 
that there must be some point between a and / whose motion is neither 
quickened nor checked by the combined action of the balls above and 
below it, and where, if a single ball were placed, it would make the same 
number of vibrations in a given time that the compound pendulum does. 
Shorten pendulum B, and find the required point. This point is called 
the center of oscillation. 

Every compound pendulum is equivalent to a simple pendulum, 
tvhose length is equal to the distance between the center of oscil- 
lation ^ and the point of suspension of the compound pendulum. 
Inasmuch as the distance between the point of suspension and 
the center of oscillation determines the rate of vibration^ 
whenever the expression length of pendulum is used it must 
be understood to mean this distance. Strictly speaking, a 
simple pendulum is a heavy material point suspended by a 
weightless thread. Of course such a pendulum cannot actually 
exist ; but the leaden ball, suspended by a thread, is a near 
approximation to it. 

Experiment ^. — From the frame (Fig. 54) suspend at one of its ends 
a lath (AB, Fig. 65) 1^ long. Find, as above, its center of oscillation.- 
It will be found to be about two-thirds the length of the lath below 
the point of suspension. Attach a weight to the lower end of AB ; 

1 The center of oscillation may be defined as that point in a pendulum at which, 
if its entire mass were collected, its time of vibration would be unchanged. 



its vibrations are now slower, and the simple pendulum B must be 
lengthened to vibrate in the same time as the lath and weight ; hence the 
center of oscillation of the lath is lowered by the addition of the 
weight. Move the weight up the lath ; the vibrations are quick- 
ened. (What is the office of a pendulum bob ?) 

Experiment 5. — Remove the weight, bore a hole through the 
lath at its center of oscillation C, and, passing a knitting-needle 
through the hole, invert the lath and suspend it by the needle. 
The pendulum is now apparently shortened, and we naturally 
expect that its vibrations will be quicker than when suspended 
from A. But the part B C now vibrates in opposition to the part 
CA, rising as it sinks, and sinking as it rises. This tends to 
check the rapidity of the vibrations of C A, and it is found that 
the pendulum vibrates in the same time when suspended from ^^^- ^^• 
C as when suspended from A. The point of suspension and the center 
of oscillation are interchangeable. 


68. Center of percussion. 

Experiment 6. — Suspend the lath by a string attached to one of its 

extremities, and with a club 
strike it horizontally near its 
upper extremity. This end of 
the lath moves in the direction 
of the stroke (A, Fig. 56), at 
the same time causing a sudden 
jerk on the string, which is felt 
by the hand. Strike the lath 
in the same direction, near its 
lower extremity ; the upper 
end of the lath now moves in 
a direction opposite to the 
stroke (B), at the same time 
causing a similar jerk of the 
string. Next strike the lath successively at points higher and higher 
above its lower extremity ; it is found that the jerk on the string becomes 
less till the center of oscillation is reached, when no pull on the string is 
felt, and neither end of the lath tends to precede the other, but both 
move on together (C). The full force of the blow is spent in moving the 
stick, and none is expended in ]3ulling the string. This point is called 
the center of percussion. 

Fig. 56. 




The center of percussion is coincide^it with the center of 
oscillation. It is the point where a blow, given or received, is 
most effective, and produces the least stress upon the support 
or axis of motion. The base-ball player soon learns at what 
point on his bat he can deal the most effective blow to the 
ball, and at the same time feel the least tingle in his hands. 

69. Demonstration of the earth'' s rotation on its axis. 

ball and string pendulum be set in vibration in a certain plane, 

then by virtue of the First Law of 
Motion it will continue to vibrate 
in the same plane even if the string 
is twirled so that the ball rotates 
on its axis. If the pendulum be 
suspended from the ceiling of a 
cabin in a vessel and the vessel be 
turned completely around, the 
plane of vibration will not be 
changed. A hammock suspended 
on deck of a ship retains the nor- 
mal position independently of the 
roll and rock of the ship, i.e. the 
hammock does not swing, but the 
ship supporting the hammock 
swings. Now if a graduated cir- 
cle be placed just beneath the ball, 
as the vessel turns about the ball 
will cross the circle at different 
points and will appear to be chang- 
ing its plane of vibration ; but it 
is evident that this appearance is 
deceptive, and that the graduated circle must turn around with the vessel, 
and that the pendulum is merely pointing out the angular motion of the 
vessel. In the same manner if a pendulum were suspended at one of the 
poles of the earth, in 24 hours every meridian of the earth would be 
brought beneath it, and although it would not meanwhile change its 
plane of vibration, it would appear to move from east to west, or in 
opposite direction to that of the earth, at the rate of 15° per hour. At 
the equator there would be no change. Between the equator and the 
pole the change per hour would vary from 0° to 15° according to the 

Fig. 57. 

earth's rotation on its axis. 83 

latitude. With a heavy metal ball and a wire as small as will support 
the ball and very long (Fig. 57), one may successfully repeat this cele- 
brated experiment, by which Foucault demonstrated the motion of the 
earth on its axis. 

Questions and Prohlems. 

1. What is the length of a pendulum that beats half-seconds? Quar- 
ter-seconds? That makes one vibration in two seconds? That makes 
two vibrations per minute ? 

2. State the proportion that will give the number of vibrations per 
minute made by a pendulum 40 cm long. 

3. Where is the center of percussion in a hammer or axe ? 

4. At what point (disregarding the length and weight of the striker's 
arm) should a blow be dealt with a bat of uniform dimensions when held 
in the hand at one extremity ? 

5. What change in the location of the center of percussion is produced 
by making one end of a bat heavier than the other ? 

6. Which end of a bat, the heavier or the lighter, should be held in 
the hands ? Why ? 

7. One pendulum is 20 inches long, and vibrates four times as fast as 
another. How long is the other ? 

8. a. What effect on the rate of vibration of a pendulum has the weight 
of its bob ? 6. What effect has the length of the arc ? c. What affects 
the rate of vibration of a pendulum ? 

9. How can you quicken the vibration of a pendulum threefold ? 

10. A clock loses- time. a. What change in the pendulum ought to 
be made ? h. How would you make the correction ? 

11. Two pendulums are four and nine feet long respectively. While 
the short one makes one vibration, how many will the long one 

12. What is the time of vibration of a pendulum (39.09 -f 4 =) 9.77 in. 

13. The number of vibrations made by a given pendulum in a given 
time varies as the square root of the force of gravity. Force of gravity 
at any place is expressed by the value of g {L e. by the acceleration which 
it produces), a. If at a certain place a pendulum 39.09 in. long make 
3600 vibrations in an hour, and the value of g be 32.16 ft., what is the 
acceleration at a place where the same pendulum makes 3590 vibrations 
in the same time ? b. Which of the two places is nearer the centroid of 
the earth ? 


14. Suggest some way by which the force of gravity at different 
latitudes and altitudes may be determined. 

15. A pebble is suspended by a thread 2 ft. long ; required the number 
of vibrations it will make in a minute. 

Section X. 


70. Work. — Whenever a force causes a change of motion 
or maintains motion against resistance it is said to do work. 
A force may act for an indefinite time without doing work ; 
for example, a person may support a stone for a time and 
become weary from the continuous application of force to 
prevent its falling, but he does no work because he effects no 
change of motion or position. A force to do work 'must effect 
a change of position. Force and &pace are essentials of work. 
Force without motion is not work ; motion without force is 
not work. The planets move, but do not work. Let the person 
supporting the stone exert a little more force, — the stone 
will rise and work will be done. An unbalanced force always 
does work. 

The body that moves another body is said to do work upon it ; 
and the body moved is said to have taork done upon it. 

When the heavy weight of a pile-driver is raised, work is 
done upon it ; when it descends and drives the pile into the 
earth, work is done upon the pile, and the pile in turn does 
work upon the matter in its path. 

71. Energy. — By the energy of a body is meant '' its 
capacity for doing work" (Maxwell). It is measured by the 
quantity of work which the body possessing it is capable of 
doing.; hence the unit of work is also the unit of energy. The 
act of doing work consists in a transfer of energy from the 
body doing work to the body on which work is done, as when 
the wind propels a vessel ; or it consists in a transformation 


of one kind of energy into another kind, as when the pile 
driver strikes the pile and the pile is forced into the earth. 
Here, a ^Dart of the energy in each act is transformed into 
heat, which we shall learn, farther on, is molecular energy. 
Work, therefore, may be defined as the act of transmitting or 
transforming energy. 

"We are acquainted with matter only as that which may have 
energy communicated to it from other matter,, and which may in its 
turn communicate energy to other matter. Energy, on the other 
hand, we know only as that which in all natural phenomena is con- 
tinually passing from one portion of matter to another. ' ' (Maxwe ll. ) 

72. Kinetic and ^potential energy. Experience teaches that 
every moving body can impart motion, therefore it can do 
work upon another ; hence every moving body possesses energy. 
The energy which a body possesses in consequence of its 
motion is called kinetic (motion) energy. It is a property of 
a moving mass only. It is capacity for doing work possessed 
by a mass in virtue of its motio7i. 

When a body is projected upward its kinetic energy dimin- 
ishes as it rises and finally becomes nil, but it is not lost, for 
it is regained as the body falls. Its energy becomes, while 
rising, stored up in virtue of its higher position. Energy in 
store, i.e. not in an active state, is called potential energy. It 
is the capacity for doing work possessed by a mass in virtue 
of its p)osition being such that it is possible for it to move, and in 
virtue of the existence of a stress tvhich tends to move it. Hence 
it is convertible into kinetic energy without the agency of any 
additional work except to remove obstacles to the conversion. 
Potential or positional energy implies force, or a tendency to 
motion, as truly as kinetic energy implies motion. 

Illustrations of energy in the potential state : 

(1) A stone lying on the ground is devoid of energy. Baise 
it and place it on a shelf ; in so doing you perform work upon 


it. As you look at it lying motionless upon the shelf, it 
appem^s as devoid of energy as when lying on the earth. 
Attach one end of a cord to it and pass it over a pulley and 
wind a portion of the cord around the shaft connected with a 
sewing-machine, lathe, or other convenient machine. Suddenly 
withdraw the shelf from beneath the stone. The stone moves ; 
it communicates motion to the machinery, and you may sew, 
turn wood, etc., with the energy given to the machine by the 

The work done on the stone or the energy transmitted to 
the stone in raising it, was not lost ; it was recovered while 
the stone was descending. There is a very important differ- 
ence between the stone lying on the ground, and the stone 
lying on the shelf : the former is powerless to do work ; the 
latter can do work. Both are alike motionless, and you can 
see no difference, except an advantage that the latter has over 
the former in having a position such that it can move. What 
gave it this advantage ? Work. A body, then, may possess 
energy due m^erely to advantage of position, derived always 
from work performed upon it. We see, then, that energy may 
exist in either of two widely different states. It may exist 
in bodies by virtue of their actual motion, or it may exist in 
bodies by virtue of their having an opportunity to move, as in 
the stone lying on the shelf. 

Possibly some will object that the work done is performed 
by gravity, and not by the stone's energy ; that if this force 
should cease to exist, the stone would not move if the shelf 
were removed, and consequently no work would be done. All 
this is very true, and it is likewise true that when the stone 
is on the ground the same force of gravity is acting, but can 
do no work simply because the position of things is such that 
the stone cannot move. The energy which the stone on the 
shelf possesses is due to the fact that its position is such that 
it can move, and that there is a stress between it and the earth 


which will cause it to move. Both advantage of position and 
stress are necessary, but the former is attained only by work 
performed. The force of gravity is employed to do work, as 
when mills are driven, by falling water ; but the water must 
first be raised from the ocean-bed to the hillside by the work 
of the sun's heat. The elastic force of springs is employed 
as a motive power : but this power is due to an advantage of 
position which the molecules of the springs have first acquired 
by work done upon them. 

We are as much accustomed to store up energy for future 
use as to store up provisions for the winter's consumption. 
We store it when we wind up the spring or weight of a clock, 
to be doled out gradually in the movements of the machinery. 
We store it when we bend the bow, condense air, or raise 
any body above the earth's surface. 

(2) Matter may possess potential energy in virtue of 
chemical separation and chemical affinity, and the potential 
energy is a measure of the work done in effecting the sepa- 
ration. For example : the entire value of coal consists in its 
potential energy, which was stored up by the work performed 
through the agency of the sun's energy in separating the 
carbon of carbon dioxide from the oxygen. Gunpowder pos- 
sesses, in a dormant state, energy sufficient to do a quantity 
of work, e.g. in blasting, which would require many laborers 
a long time to do. 

A body possesses 'potential energy when, in virtue of work done 
upon it, it occupies a position of oAimntage, or its constituent 
particles occupy positions of advantage, so that the energy ex- 
pended caM he at any time recovered hy the return of the body to 
its original position, or by the return of its particles to their 
original positions. 

73. Relation of energy to force and matter. — Our discussions lead 
us to conclude that energy is a condition of matter, due either to its 
motion or to its relation with other matter, in virtue of which the matter 


is capable of doing work. Force i may be regarded as ' ' the measure 
of the tendency of energy" to transfer or "to transform itself." 

Energy is never found except in association with matter. Hence 
matter may be defined as the veliicle or receptacle of energy. The 
First Law of Motion affirms that matter is simply passive, inert. 

74. Fractical imits of work and energy. — Inasmuch as a 
body's capacity to do work is dependent wholly upon the work 
which has been done upon it^ it is evident that both work and 
energy may be measured by the same unit. The practical 
unit adopted is the work done or energy irri'parted in raising one 
•pound through a vertical hight of one foot. It is called a foot- 
pound. The metric unit is the work done or energy imparted 
in raising 1 K a vertical hight of 1 m, and is called a kilogram- 
meter. The kilogrammeter is equivalent to 7,2331 ft. lbs. 
Since the work done in raising 1 pound 1 foot high is 1 foot- 
pound, the work of raising 1 pound 10 feet high is 10 foot- 
pounds, which is the same as the work done in raising 10 
pounds 1 foot high ; and the same, again, as raising 2 pounds 
5 feet high. 

There are many kinds of work besides that of raising 
weights. But since, with the same resistance, the work of 
producing motion in any other direction is just the same as 
in a vertical direction, it is easy, in all cases in which the 
resistance and space through which the resistance is overcome 
are known, to find the equivalent in work done in raising a 
weight vertically. By thus securing a common standard for 
measurement of work, we are able to compare any species of 
work with any other. For instance, let us compare the work 
done in sawing through a stick of wood by a man whose saw 

1 •■' By a convenient form of speech a given force is said to act upon a given body 
and to impart to it a given acceleration. It must be constantly borne in mind, how- 
ever, that a force is not a physical entity, and can never be measured until we 
already know, absolutely or by comparison, the mass acted upon and the acceleration 
imparted to it." (Danteli,.) " Force is a mere phantom suggestion of our muscular 
sense." (Tait.) " Energy has its price, force has not." 



must move 10 m against an average resistance of 12 K, with 
that done by a bullet in penetrating a plank to a depth of 
2 cm against an average resistance of 200 K. Moving a saw 
10 m against 12 K resistance is equivalent to raising 12 K 
mass 10 m high, or doing 120 kgm of work ; a bullet moving 
2 cm against 200 K resistance does as much work as is re- 
quired to raise 200 K mass 2 cm high, or 200 X .02^4 kgm 
of work. 120 -H 4 = 30 times as much work done by the 
sawyer as by the bullet. 

75. Absolute units of work} — If force be measured in dynes, 


Fig. 58. 

and distance in centimeters, the work done is expressed in a 
C.G.S. unit called an erg. An erg is the luork done or energy 
imparted by a force of one dyne working through a distance of 
one centimeter. 

In purely scientific investigations absolute units are em- 

1 The pupil will, perhaps, be assisted by the accompanying diagram (Fig. 58) in 
his first attempts to acctuire and classify the units of force, energy, and work in the 
several systems. 


The following equivalents will be useful : 

Gravitation units absolute units 

1 gram-centimeter =c= g ergs, 

1 kilogram meter =c= 100^000 g ergs, 

1 foot-pound =c=: 0.13825 X 10^ X ^ ergs. 

76. Formulas for calculating luork or energy imparted. — 
Force and space (ov distance), being essentials of work (p. 84), 
are necessarily the quantities employed in calculating work. 
A given force acting through a space of one foot does a certain 
quantity of work ; it is evident that the same force acting 
through a space of two feet would do twice as much work. 
Hence the general formula 

W=fi, (1) 

in which / represents the force employed, s the space through 
which the force acts, and W the work done. 

In case a force encounters resistance, the magnitude of the 
force necessary to produce motion varies with the resistance 
(Third Law of Motion). Often the work done upon a body is 
more conveniently determined by niultijolying the resistance hy 
the space through luhich it is overcome, and our formula becomes 
by substitution of r (resistance) for / (the force which over- 
comes it) 

rs=W: (2) 

For example, a ball is shot vertically upward from a rifle in a 
vacuum ; the work done upon the ball (by the explosive force 
of the gunpowder) may be calculated by multiplying the 
average force (difficult to ascertain) exerted upon it, by the 
space through which the force acts (a little greater than the 
length of the barrel); or by multiplying the resistance to 
motion offered by gravity, i.e. its weight (easily ascertained), 
by the distance the ball ascends. 


Let us calculate the energy stored in a bow by an archer 
whose hand, in bending the bow by pulling on the string, 
moves 6 inches (^ foot) against an average resistance of 
20 pounds. Here rs ^20 X i^= 10 foot-pounds of work done 
upon the bow, or 10 foot-pounds of energy stored in the bow. 

77. Formula for calculating kinetic energy. — Suppose a body 
to have a mass m and a velocity v ; it can do a definite 
quantity of work before it is thereby brought to rest. If it 
be moving upward a mutual work between it and the earth is 
performed in destroying each other's momenta. If its velocity 
be such that it will rise to a hight s, then its kinetic energy 
is such that it will do m g s absolute units of work, or 

E]c (Kinetic energy) ^ m g s. (1) 

We may find, then, to what vertical hight a body having 
a given velocity would rise if directed upward, and from 
formula (1) determine its kinetic energy ; but a formula may 
be obtained which will give the same result with less trouble ; 
thus, substituting g for a in the formula v = at (page 10), we 
have v=-gt\ whence 


t=-- , or t~ = —^- 


Again 5 = -^^/?^^; substituting the value of t'^ in this equation 
we have 

g ^g 

Substituting for s in equation (1) its value we have 

^^^ = ^' (2) 

a formula which will determine the kinetic energy of a body 
in absolute units when its mass and velocity are known, since 
the energy is the same whatever be the direction of the 


Hence the kinetic energy of a body is half the product of its 
mass by the square of its velocity. 

If the result be desired in gravitation units, i.e. in gram- 
centimeters or foot-pounds, the number of absolute units must 
be divided by g^ since g ergs (980) are equivalent to one 
gram-centimeter, or g foot-poundals (32.2) are equivalent to 
one foot-pound. 

Work done by a force is measured by the product of the numeric 
of the force and that of the space s or [L] through which it acts. 
Then since force [/] = [MLT— 2]^ the dimensional formula for 
work [W] or energy [E] is, therefore, {fs=) [ML^T-^]. 

78. Energy co7itrasted luith momentum.'^ — It is evident from 
formula (2) that when the mass (m) of a body remahis the 
same, its energy is pvojiortional to the square of its velocity ; 
while its momentum, as we have learned, is proportional to its 
velocity. In other words, the effect of increasing the velocity 
of a moving body would seem to be to increase its working 
power much more rapidly than its momentum. Is this py^ac- 
tically true ? 

Experiment. — Fill a water-pail with moist clay. Let a leaden bullet 
drop upon the clay from a hight of .5 m. Then drop the same bullet 

1 Problem. — A bullet weighing 30 g is shot with a velocity of 98 m per second from 

a gun weighing 4 K ; required the momentum and the energy of both the bullet and 

the gun, and the velocity of the gnn. Solution: Using the kilogram, the meter, and 

the second as units, the momentum of the ball is .03 x 98 = 2.94 units. If the ball 

were shot vertically upward, its velocity would diminish 9.8 m per second ; so it would 

rise ^ = 10 seconds, and, therefore, before its energy was expended, to a hight of 

(§ 13) 4.9 m X 102 = 490 m. Hence, its energy at the outset is .03 x 490 = 14.7 kgm. By 
the third law of motion the momentum of the gun must be just the same as that of 
the ball, 2.94 units; its velocity is therefore 2.94^.4= .735m per second. Then 

t = '-—— = .075 second ; the hight (supposing the gun to be raised vertically by the 

impulse received) = 4.9 x .0752 = .02766 m ; and its energy = 4 x .02766 = .1102 kgm. 

While, therefore, the momenta generated in the two bodies by the burning of the 

14 7 
powder are equal, the energy of the bullet is ' = 133^ times that of the gun. 

(Why are the effects produced by the bullet more disastrous than those produced by 
the recoil of the gun ?) 


from a hight of 2 m, or four times the former bight, in order that it may 
acquire twice the velocity. In the latter case it penetrates to four times 
the depth that it did in the former. 

So it appears that the energy of a moving body varies, not as 
its velocity, but as the square of its velocity. Doubling the 
velocity increases the energy fourfold, trebling the velocity 
increases it ninefold, and so on ; but the corresponding 
momentum is increased only twofold, threefold, etc. A bullet 
moving with a velocity of 400 feet per second will penetrate, 
not twice, but four times, as far into a plank as one having a 
velocity of 200 feet per second. A railway train having a 
velocity of 20 miles an hour will, if the steam be shut off, 
continue to run four times as far as it would if its velocity 
were 10 miles an hour. The reason is now apparent why 
light substances, even so light as air, exhibit great energy 
when their velocity is great. 

Furthermore we have seen, p. 37, that momentum = 71^ F 
=ft ; and again, p. 90, W (work done) or Ek (kinetic 
energy imparted) ^= fs. 

The momentum, then, imparted to a body is the product of 
the force into the time it acts ; energy imparted is the product 
of force into the space through which it acts. It is evident, 
therefore, that force may be measured by the momentuim when 
time is considered, and by the energy which it imparts when the 
space is considered. 

Questions and Problems. 

1. Does the energy expended in raising the stones to their places in 
the Egyptian pyramids still reside in the stones ? 

2. What kind of energy is that contained in gunpowder ? 

3. Can a person lift himself, or put himself in motion, without 
exerting force upon some other body ? 

4. a. Can a body do work upon itself ? b. Can a body generate 
energy in itself, i. e. increase its own energy ? 


5. a. Suppose that an average force of 25 pounds is exerted through 
a space of 10 inches in bending a bow ; what amount of energy will it 
give the bow ? b. What kind of energy will the bow, when bent, 
possess ? 

6. a. What amount of kinetic energy does a mass of 20 pounds moving 
with a velocity of 300 feet per second, possess ? b. What amount of work 
can the body do ? 

7. How many fold is the kinetic energy of a body increased by 
doubling its velocity ? 

8. How high will twelve hundred foot-pounds of energy raise 100 
pounds ? 

9. A force of 500 pounds acts upon a body through a space of 20 feet. 
One-fourth of the work is wasted in consequence of resistances. How 
much available energy is imparted to the body ? 

10. How much energy is stored in a body weighing 1,000 pounds, at a 
bight of 200 feet above the earth ? 

11. A horse draws a carriage on a level road at the uniform rate of 
5 miles an hour. a. Does work accumulate ? b. What kind of energy 
does the carriage possess ? c. Suppose that the carriage were drawn up 
a hill ; would energy accumulate ? d. What kind of energy would it 
possess when at rest on the top of the hill ? e. How would you calculate 
the quantity of energy it possesses when at rest on top of the hill ? 
/. Suppose that the carriage is in motion on top of the hill ; what two 
formulas would you employ in calculating the total energy which it 
possesses ? 

12. How much work is done per hour if 80 K be raised 4 m per 
minute ? 

13. a. What energy must be imparted to a body weighing 50 g 
that it may ascend 4 seconds ? b. How many times as much energy 
must be imparted to the same body that it may ascend 5 seconds? 
c. Why? 

14. Compare the momenta, in the two cases given in the last question, 
at the instants the body is thrown. 

15. How much energy is stored in a body which weighs 50 K, at a 
hight of 80 m above the earth's surface ? 

16. How much kinetic energy would the same body have if it had a 
velocity of 100 m per second ? 

17. Suppose it to fall in a vacuum, how much kinetic energy would it 
have at the end of the fourth second ? 

18. If it should fall through the air, what would become of a part of 
the energy ? 


19. A projectile of mass 25 K is thrown vertically upward with an 
initial velocity of 29.4 m per second. How much energy has it ? 

20. What becomes of its energy during its ascent ? 

21. a. Compare the momentum of a mass 50 K having a velocity of 
2 m per second, with the momentum of a body of a mass 50 g having 
a velocity of 100 m per second, h. Compare their energies. 

22. Which, momentum or energy, will enable one to determine the 
amount of resistance that a moving body may overcome ? 

23. Explain how a child who cannot lift 30 K can draw a carriage 
weighing 150 K. 

24. How many and what transformations take place during a single 
swing of a pendulum ? 

25. What quantity of energy will be expended if a force of 60 lbs. 
move a body a distance of 20 ft. ? 

26. A body of mass 30 lbs. moving with a velocity of 50 ft. per sec. 
must do how much work before it stops ? 

27. A United States 12-inch army gun, using a charge of 440 lbs. of 
powder, throws a projectile of mass 1000 lbs. with an initial velocity of 
1975 ft. per second, a. What quantity of energy is imparted to the 
projectile ? h. The maximum range of this gun is 15 miles. If the 
velocity of the projectile were uniform, in how many seconds would it 
strike the ground ? c. If the projectile were directed vertically upward, 
during how many seconds would it rise, the resistance of the air being 
disregarded ? d. How high would it rise ? 

28. a. A street car having a mass of 3 tons was moving at the rate of 
6 miles per hour. The brakes were applied and stopped it in 4 seconds. 
What was the average force exerted by the brakes ? 6. Suppose the car 
to have been stopped in the space of 40 ft. , what was the average force 
applied ? 

79. Activity. — The activity (sometimes called the power) of 
an agent is the rate at which it does or can do work ; or it is 
the quantity of work it does in a unit of time, and is deter- 
mined by the formula 

. , ^. .^ - W (work) 

In estimating the total quantity of work done, the time 
consumed is not taken into consideration. The work done by 


a hod-carrier in carrying 1000 bricks to the top of a building 
is the same whether he does it in a day or a week. But in 
estimating activity, as of a man, a horse, or a steam-engine, 
in other words, the rate at which the agent is capable of doing 
work, it is evident that time is an important element. The 
work done by a horse in raising a barrel of flour 20 feet is 
about 4000 foot-pounds ; but even a mouse could do the same 
quantity of work in time. 

The unit in which activity or rate of doing work is estimated 
is called (inappropriately) a horse-power. A horse-power is 
550 foot-pounds per second, 33,000 foot-pounds per minute, 
or 1,980,000 foot-pounds per hour. 

The logical unit of activity is a unit of work in a unit of 
time, as one erg per second. The absolute unit of activity, 
chiefly used in measuring electrical activity, is the loatt, or 
ten million ergs per second. A moderate estimate of man's 
activity is 100 watts. 

1 erg per second -z- .0000001 watt, 

1 horse-power =r= 746 watts, 

1 foot-pound per minute -z- 226,043 ergs per second. 

Activity being measured by the numeric of the work done per 
unit of time, its dimensional equation is therefore [A] = [WT~i] 
= [ML2T-3]. 

Questions and Problems. 

1. For which is a truck- liorse valued, his energy or his activity ? 

2. Do we speak of the activity or the energy of a steam-engine ? 

3. Which do we apply to levers and machines in general, povv^er or 
force ? 

4. ' ' Energy is the power of doing work, ' ' Is this true ? 

5. Shall we say that the activity, or the energy, of the horse is greater 
than that of man ? 

6. How much work can a 2 horse-power engine do in an hour ? 


7. a. What quantity of work is required to raise 50 tons of coal from 
a mine 200 feet deep ? b. An engine of how many horse-power would be 
required to do it in two hours ? 

8. A car of mass 6000 k is drawn by a horse at a speed of 100 m per 
minute. The index of the dynamometer to which the horse is attached 
stands at 40 K. a. At what rate is the horse working ? b. Express the 
rate in horse-power. 

9. A dynamometer shows that a span of horses pull a plow with a 
constant force of 70 K. What activity is required to work the plow if they 
travel at the rate of 3 km per hour ? 

10. What horse-power in an engine will raise 1,850,000 K 5m in an 
hour ? 

11. How long will it take a 3 horse-power engine to raise 10 tons 
50 feet ? 

12. How far will a 2 horse-power engine raise 1000 K in 10 seconds ? 

13. A force of 10,000 dynes acting through a space of 100 meters per 
second furnishes an activity of how many watts ? 

14. The wind moves a vessel with a uniform velocity of 5 miles an 
hour against a constant resistance of 2000 pounds. What activity is 
furnished by the wind ? 

15. If a 2 horse-power engine can just throw 1056 pounds of water to 
the top of a steeple in two minutes, what is the hight of the steeple ? 

16. A cannon ball of 10,000 g is discharged with a velocity of 45,000 cm 
per second. Find its kinetic energy. Ans. 10125 X 10^ ergs. 

17. In the last question, find the mean force exerted upon the ball by 
the powder, the length of the barrel being 200 cm. 

Ans. 50625 X 10^ dynes. 

18. Supply the following ellipses by selecting appropriate words from 
the following : viz. force, work, energy, activity. When — acts through 
space — is performed, and — is imparted. The rate at which — is per- 
formed determines the — of the agent. The — of a bullet flying through 
vacant space. The — of a horse.- The ■ — of wind. The — of a bent 
bow. What — must a bullet of mass 1 ounce have that it may rise 4 
seconds ? What — is consumed by a steamer in crossing the ocean ? 
What — is necessary that it may traverse 300 knots per day, and what 
must be the average — exerted to overcome the resistances at the required 



Section XI. 


80. Uses of maclmies. 
Experiment 1. — Suspend, 

as in Fig. 59, a fixed pulley, A, and a 
movable pulley, B. Let the scale-pan 
C counterbalance the pulley B, so that 
there will be equilibrium . Suspend from 
B two balls, LL, of equal weight, and 
suspend on the side where the pan is, 
a single ball, K, equal to one of the 
former. The single ball supports the 
two balls; i.e. by the use of the machine, 
a force of 1 is enabled to balance a force 
of 2. So far no work is done. Place a 
very small weight in the pan ; the balls 
LL rise, and work is done. 

As the weight K plus a very small 
weight causes the motion, we shall 
regard this as the force (/); and as the 
weights LL are the bodies moved (the 
pulleys and pan being parts of the 
machine may be disregarded), they may 
be regarded as the resistance (r) over- 
come, or the body on which work is 
done. Measure the respective distances through which / acts and r moves 
during the same time, r moves only one-half as great a distance as that 
through which / acts; i.e. if r rise 2 feet, / must act through 4 feet. 
Suppose that r is 2 pounds, then / is 1 + pounds. Now 2 (pounds) X 2 
(feet) = 4 foot-pounds of work done on r. Again, 1 + (pounds) X 4 (feet) 
= a little more than 4 foot-pounds of work (or energy) expended. 

It thus seems that, although a machine will enable a small 
force to balance a large force, when work is performed the 
work applied to the machine is greater, rather than less, than 
the work which the machine transmits to the resistance. The 
work applied is greater than the work transmitted by the 
amount of work wasted in consequence of friction and other 


resistances. So that hy the employment of a machine nothing 
is gained in work, hnt something is aliuays lost. 

What, then, is the advantage gained in using this machine? 
Suppose that r is 400 pounds, and that the utmost force that 
a man can exert is a little more than 200 pounds. Then 
without the machine the services of two men would be required 
to move the resistance ; whereas one man can move it with a 
machine, but he will be obliged to move twice as far as the 
resistance moves, a matter of little consequence in comparison 
with the advantage of being able to do the work alone. The 
advantage gained in this instance seems to be one of convenience. 
Men, however, are accustomed to speak of it as "ct gain of 
force ^^ (or more commonly and inaccurately, ^^of poiuer^^), 
inasmuch as a small force overcomes a large resistance. 

Experiment 2. — If instead of applying the small additional weight to 
the pan, it he suspended from one of the balls LL, the weight of these 
balls, together with the additional weight, becomes the cause of motion, 
and K is the resistance. In this case there is a loss of force, because the 
force employed is greater than the force overcome. Measure the dis- 
tances traversed respectively by K and LL in the same time. K moves 
twice as far as LL, and of course with twice the speed. There is a gain 
of speed at the expense oi force. 

It thus appears that, if it should be desirable to move a 
resistance with greater speed than it is possible or con- 
venient for the force to act, it may be accomplished through 
the mediation of a machine, by applying to it a force propor- 
tionately greater than the resistance. This apparatus is one 
of many conti'ivances called machines, through the mediation of 
which force can he applied to resistance more advantageously 
than when it is applied directly to the resistance. 

At present we deal with machines employed as means for 
transmitting and modifying motion and force. Later we shall 
consider machines whose function is to transform energy, such 
as the steam engine, dynamo, etc. 



Some of the many advantages derived from the use of 
machines are : 

(1) They may enable us to exchange intensity of force for 
speed, or s])eed for intensity of force. A gain of intensity of 
force or a gain of speed is called a mechanical advantage. 

(2) They may enable us to emi^loy a force in a direction 
that is more convenient than the direction in which the resis- 
tance is to be moved. 

(3) They may enable us to employ other forces 
than our own muscular force in doing work; 
e.g. the muscular force of animals ; the forces 
of wind, water, steam, etc. 

How are the last two uses illustrated in 
Fig. 60? The pulleys employed are called 
fixed pulleys, i.e. they have no motion except 
that of rotation. Is any mechanical advantage 
gained by fixed pulleys ? What is the use of a 
fixed pulley ? Pulley B (Fig. 59) is a movable 
pulley. What advantage is gained by means of 
a movable pulley ? 

81. General law of machines. — 
From the experiments and discussion 
above we derive the following for- 
mula for machines : 

fs = rs' -\r w, (1) 

in which / represents the force applied, and s the distance 
through which / acts ; r represents the resistance overcome, 
and s' the distance through which its point of application is 
moved ; lu represents the wasted work. A machine in which 
there is no wasted work is a perfect machine. Such a machine 
is purely ideal, as none exists. If in our calculations we 
regard a machine as perfect (though subsequently suitable 
allowance must be made for the wasted work), then our 
formula becomes fs = rs', (2) 

Fig. 60. 


whence r :/:*: s : s'; i.e. the force and resistance vary inversely 
as the distances which their respective points of application move. 
In other words, the ratio of the resistance to the force is the 
reciprocal of the ratio of the distances which these points 
move ; thus, if 

r : f:= 4, then s' : s = J. 

This law applies to machines of every description; hence it is 
called the General or Universal Law of Machines. When r 
is greater than /, there is a gain of intensity of force, and 


- = the ratio of gain of intensity of force. When s' is greater 

than s, there is a gain of speed, and - =: the ratio of gain of 


Since fs^ the work done upon a machine, is always greater 
than rs', the work transmitted by the machine, we infer that 
no machine creates or increases energy. No machine transmits 
more energy than it receives. A machine may enable us to 
gain intensity of force, but not energy. By taking s great 
enough,. / can be made as small as we please ; in this case in 
proportion as force is gained, time, distance, or speed is lost. 

Formula (1) exhibits the relation between the work performed 
upon the machine and the entire work transmitted and transformed 
by the machine. But at any given instant while the machine is 
in operation, it is evident that (see p. 91) 

/s = i mv'^ + rs' + w. (3) 

In formula (3) \mv'^ represents the kinetic energy of the moving 
parts of the machine. 

82. Efficiency of machines. — The efficiency of a machine is 
a fraction, usually a per cent, expressing the ratio of the 
energy given out by the machine and utilized to the total 
energy expended upon the machine. The limit of the efficiency 
of a machine is unity, which is the efficiency of an " ideal," 


or perfect, machine, in which no energy is lost. The object 
of improvements in machines is to bring their efficiency as 
near to unity as possible. For instance, if 50 foot-pounds of 
energy be expended on a machine, and friction convert 8 foot- 
pounds into heat and 5 foot-pounds be lost in consequence of 
only a component of the working force being utilized, so that 
the machine is able to give out only 37 foot-pounds, its 
efficiency is f ^ = 74 per cent. If the friction can be reduced 
one half and an improvement can be made in the machine 
which will render the entire working force effective, then 
there will be wasted only 4 foot-pounds of energy, and its 
efficiency will be raised to f ^ = 92 per cent., and the quantity 
of work which the machine will accomplish will be increased 
in the ratio of 92 : 74. 

83. Mechanical j^oivers. — Machines, however complicated 
or complex, are largely composed of a few simple machines 
long known as the ''mechanical poiuersJ' As usually given 
they are the Lever, Wheel and Axle, Inclined Plane, Wedge, 
Screw, Pulley, and Knee. 

84. Experiments luith the lever. 

Experiment 3. — Support a lever, as in Fig. 61, so that there shall be 
unequal arms. Move W until the lever is balanced in a horizontal posi- 
tion. Suspend (say) seven balls from the short arm (say) one space 

from the fulcrum. Then from 

.^^ i _^,___^_^_ ^_^oy . ^ i^ ^ '^ \-^_j ^j^ the other arm suspend a single 

' (|) ball from such a place (in this 

ife case seven equal spaces from the 

) Q fulcrum) that it will balance the 

Q seven balls. There is now equili- 

FiG. 61. ' ' ' brium between the two forces. 

Suppose the smaller force to be 

increased a little and to produce motion; what mechanical advantage 

{i.e. intensity of force or speed) would be gained by the use of the 

machine ? What is the ratio of gain, the small additional force being 

neglected ? How does this ratio compare with the ratio between the 

length of the two arms ? For convenience we call the distance of the 




point of application of tlie force from the fulcrum the force-arm^ and the 
distance of the resistance from the fulcrum the resistance-arm. 

Suppose the small additional force to he applied to the short arm ; 
what mechanical advantage would be gained ? What would be the 
ratio of gain ? 

While the general law of machines (p. 100) is always 
applicable, its application is not always convenient, since, for 
example, it necessitates putting the machine in motion in 
order to measure s and s' 
(the distances traversed 
respectively by the points 
of application of the force 
and resistance in the same 
time), an operation which 
would be very difficult 
and tedious in many 
cases. Hence, a sidecial 
law, one in which the 
relation between the ratio 
of gain and the ratio be- 
tween certain dimensions 
of the machine is stated, 
is often more convenient 
in practice. For example, 
in our experiment with 
the lever we discover that 
E, : F : : force-arm: resistance-arm, i.e. the force and resistance 
vary inversely as the lengths of their respective arms. Compare 
this special law with the general law. Place the fulcrum at 
other points in the lever, and thereby vary the length of the 
arms, and verify by numerous experiments the special law 
of levers. 

Experiment 4. — By means of a pulley, D, so arrange (Fig. 62) that 
both F and R may be on the same side of the fulcrum. First, place in 


Fig. 62. 



the pan weights sufficient to produce equilibrium in the machine (for 
example, in this case, one ball). Then suspend weights at some point, as 
A, and place other weights in the pan to coiinterbalance these. Verify 
the law of levers. If A be the resistance, what mechanical advantage is 
gained ? What is the ratio of gain ? If B be the resistance, what 
mechanical advantage is gained ? 

85. Wheel and axle. — The wheel and axle consists of two 
cylinders having a common axis, the larger of which is called 

Fig. 63. 

Fig. 64. 

the wheel, and the smaller the axle, as A and C (Fig. 63). 
The wheel may be moved by the hand or by a string with a 
weight attached to it. 

The wheel is often replaced by a crank, as in the windlass, 
or by a spoke, as in the capstan, and is thus employed in 
hoisting apparatus, such as cranes, derricks, etc. 

The wheel and axle viewed in section will be seen (Fig. 64) 
to be only a modification of the lever which, unlike the latter, 
may be continuous in its operation. C is the fulcrum, the 
radius CA is the force-arm, and the radius CB the resistance- 


arm. The laws pertaining to this machine are virtually the 
same as those of the lever. For example, when the force / is 
applied to the wheel and the resistance r is at the axle, 

R (radius of wheel) K (radius of axle) 
1 1 

C (circumference of wheel) C (circumference of axle). 

86. Inclined plane. — Any plane surface not horizontal or 
vertical, known as an inclined plane, may be used as a simple 
machine for gaining intensity of force ; e.g. a plank resting 
with one end on a cart body and the other on the ground, a 
hill-side, or a road-grade. The gradient is the quantity of rise 
per horizontal foot, or it is the ratio of the vertical rise to the 
horizontal distance. 

When a body is pressed against a hard, smooth surface, the 
resistance offered by the surface is at right angles to the 
surface. A body, e.g. a sphere, may be supported on a 
horizontal surface, for the weight acting downward is counter- 
acted by the upward reaction of the plane. But since on an 
inclined plane the reaction is not vertically upward, a body 
cannot rest on it without the aid of another force. 

The mechanical advantage of this machine depends on the 
principle of the resolution of a force into its components. Let 

AB (Fig. ^^) be an inclined plane whose gradient is — ;^- Let 

a be the centroid of the weight W (technically called the load). 
The line of direction of the load is along the vertical ao, but 
the pressure exerted upon the plane is in the direction ac, and 
the reaction of the plane is in the direction ca. We may take 
any length along the vertical as a 5 to represent the load W. 
Draw he parallel to AB to meet ac. Complete the paral- 
lelogram adhc with a^ as its diagonal. The force ah \& 
thereby resolved into two forces, ac representing the pressure 



upon the plane, and ad representing an unbalanced force 
tending to move W along the plane B A. It is evident that 
to produce equilibrium, i.e. to support the body on the plane 
AB, a force equal to ad, but opposite in direction, must be 

Fig. 65. 

employed. Now it may be proved geometrically that the 
triangles adh and BCA are similar; i.e. 

da: ah : : CB : BA. 

But da represents the force /necessary to produce equilibrium, 
while 0^^ represents the load or resistance ?-; CB represents 
the hight h, and BA the length I, of the inclined plane. 

f-.T-.-.h: I ovf=-r. 

Hence a given force acting parallel ivith the direction of 
inclination of an inclined plane, will support a weight as many 
times greater than itself as the length of the inclined plane is 
greater than its vertical hight. Corollary : with a given length 
of inclined plane the greater its vertical hight, i.e. the steeper 
it is, the greater /must be. 



Suppose that a (Fig. ^^), the point of application of r, be 
moved to V, how high is the load raised? Through what 
distance does / act ? Show that the general law of machines 
is applicable to this case. 

87. Angle of repose. 

When work is done, i.e. when the load is moved up the inclined 
plane, a force greater than / must be employed, partly to overcome 
friction, and partly to produce acceleration. On the other hand, if 
a body be allowed to roll or slide down an inclined plane, the force 
of gravitation overcomes friction and produces acceleration. Since / 
diminishes with the angle of inclination of the plane, there must be 
an angle at which / will be just equal to the friction. This angle is 
called the angle of repose. If the angle of inclination be greater 
than the angle of repose, the body is acted on by an unbalanced 
force whose magnitude is determined by the inclination of the plane, 
and the acceleration produced thereby will be proportional to the 
unbalanced force. This suggests a way of " diluting " the action of 
gravity, so as to be able to study the effects of a constant unbalanced 
force, and a marble rolling down a smooth plane very slightly 
inclined furnishes a very simple substitute for the falling weights 
in Atwood's Machine. 

88. TJie wedge is a triangular prism, used commonly as a 
movable inclined pl^ne for moving great resist- ., 
ances through short distances. It usually 
consists of two inclined planes as AcP and ^^ 
BcP (Fig. QQ). The force applied acts in the 
direction po and the resistance acts at right 
angles to the planes, or in the directions Dm 
and En. The force applied is of the nature 
of a percussion, as that of a sledge ; besides, 
friction and other resistances form so con- 
siderable a factor in its use that no definite 
law of any practical value can be given, 
further than that, with a given thickness, fig. m. 
the longer the wedge the greater the gain in intensity of 



89. The screw is another variety of the inclined plane, as 
may be shown by winding a triangular piece of paper around 
a cylinder, e.g. a lead pencil (Fig. 67). The hypotenuse will 

Fig. 67. 

Fig. 68. 

form a spiral about the cylinder resembling the threads of 
a screw. 

In actual practice the screw consists oT two parts : (1) a 
convex grooved cylinder, or scretv, S (Fig. 6S), which turns 
within (2) a hollow cylinder, or 7iict, N. The concave surface 
of the latter is cut with a thread corresponding 
to the thread of the screw. The force is em- 
ployed either to turn the screw within an immov- 
able nut, or to turn the nut about a fixed screw. 
In either case the force is usually applied to a 
lever or wheel fitted either to the screw or to 
the nut. 

During a single rotation of the screw or nut, 
the load or resistance is moved a distance equal 
to the vertical distance between the correspond- 
ing surfaces of two successive threads, usually termed the 

Fig. 69. 



pitch of the screw, as ah (Fig. 69). Then in conformity to 
the universal law of machines the force is to the resista7ice as 
the distance hettveen the corresponding surfaces of two successive 
threads is to the circumference of the circle described hy the 

Since a screw turning in a nut advances only its pitch distance at 
each revolution, a finely cut ^-^ 

screw furnishes an instru- q f QS\ ^ 

ment well adapted to meas- 
ure very minute distances. 
For example, if the screw C 
of the micrometer caliper 
(Fig. 70) have a pitch of one 
millimeter, and the thimble 
D be divided near its end A 
into one hundred parts so as 
to register hundredths of a 

revolution, it is evident that any object {e.g. a hair) placed in its 
jaws at B can be measured to the hundredth of a millimeter. 

90. The Jaiee or toggle-joi7it. — This machine also is em- 
ployed where great pressure has 
to be exerted through a small 
space, as in punching and shear- 
ing iron. Force applied at C 
(Fig. 71) in the direction CD 
will cause the jointed bars to be 
brought into line with each other 
and tend to push the objects at 
the other extremities of the bars 
apart. The nearer these bars 
approach to a straight line, the 
greater pressure will a given 
Fig. 71. force produce. 




1. a. When is a machine said to gain intensity of force? 6. When 
is it said to gain speed ? 

2. a. How is intensity of force gained by the use of a machine? 
h. How is speed gained by the use of a machine ? 

3. a. What is mechanical advantage? h. Give a rule by which the 

Fig. 72. 

mechanical advantage that may be gained by any machine may be 

4. Energy is applied to a machine at the rate of 250 ft. lbs. per minute 
and it transmits 200 ft. lbs. per minute. What is its efficiency ? 

5. Fig. 72 represents a pile-driver, a. How can the energy or the 
work which the weight W can do when it is raised a given distance be 
computed ? h. What benefit is derived from the use of the machine in 



raising the weiglit ? c. Suggest some simple attachment to the machine 
which would enable one man to raise the weight, d. Suggest some 
attachment by means of which a horse could be employed to do the 
work. e. What difference will it make whether the weight is raised 

5 feet or 10 feet ? /. Illustrate, by means of this machine, what you 
understand by force and energy. 

6. a. What advantage is gained by a nut-cracker (Fig. 73) ? b. What 
is the ratio of gain ? 

7. a. What advantage is gained by cutting far back on the blades of 
shears near the fulcrum (Fig. 74) ? "*Why ? 6. Should shears for cutting 

Fig. 74. 

metals be made with short handles and long blades, or the reverse ? 
c. What is the advantage of long blades ? 

8. The arm is raised by the contraction (shortening by muscular 
force) of the muscle A (Fig. 75), 
which is attached at one extremity 
to the shoulder and at the other ex- 
tremity B to the fore-arm, near the 
elbow, a. When the arm is used, as 
represented in the figure, to raise a 
weight, what kind of machine is it ? 
6. What mechanical advantage is 
gained by it ? c. How can the ratio 
of gain be computed ? d. For which 
purpose is the arm adapted, to gain 
intensity of force or speed ? 

9. Is work done when the moment of the force applied to a lever is 
equal to the moment of the resistance ? Why ? 

Fig. 75. 



Fig. 76. 

A 10. If r (Fig. 76) be the fulcrum of 

the lever C A, and A B represent, on a 

■ scale of 1 cm = 1 k, a force applied at A, 

what force applied at C in the direction 

C D will produce equilibrium ? 

11. With a wheel and axle a force 
of 8 lbs. sustains a weight of 56 lbs. ; what is the ratio between the radii 
of the wheel and of its axle ? 

12. A capstan turned by two horses is used to draw a boat ; the horses 
are attached to the levers 12 feet 

from the axis of the capstan ; the 
radius of the axle is 18 in. When 
each horse pulls with a force of 
1,000 lbs. what force is exerted 
upon the boat ? 

13. Suppose the screw in the 
letter-press (Fig. 77) to advance 
inch at each revolution, and a force 
of 25 pounds to be applied to the cir- 
cumference of the wheel &, whose 
diameter is 14 inches. What pres- 
sure would be exerted on articles 
placed beneath the screw ? 

14. A lever is 75 cm long ; where must the fulcrum be placed in order 
that a force of 2 k at one end may balance 4 k at the other end ? What 
will be the pressure on the prop ? 

Fig. 77. 

Fig. 78. 

15. Two weights, of 5 k and 20 k, are suspended from the ends of a 
lever 70 cm long. Where must the fulcrum be placed that they may 
balance ? 



16. If P (Fig. 78), weighing lib., is suspended 15 spaces from the 
fulcrum of the steelyard, what weight (W) suspended 3 similar spaces the 
other side of the fulcrum will balance it ? 

17. How would you weigh out 6 pounds of tea with the same steel- 
yard ? 

18. a. A skid 12 feet long rests with one end on a cart at a hight of 
3 feet from the ground. What force will roll a barrel of iiour weighing 
200 lbs. over the skid into the cart ? h. What amount of work will be 
required ? 

19. a. Draw a line to represent an inclined plane. Find what is the 
least force that will prevent a ball weighing 96 lbs. from rolling down the 
plane. 6. Find the pressure which the ball will exert upon the plane. 

20. An iron safe on trucks, weighing two tons, is prevented from 
rolling down an inclined plane by a force of 250 lbs. What is the ratio 
of the length of the plane to its hight ? 

21. The gradient of an inclined plane is 1 ft. in 4 ft. To produce 
equilibrium on this plane what relation must the force applied parallel 
with the plane bear to the load ? 

22. If the circumference of 
an axle (Fig. 79) be 60 cm, and 
the force applied to the crank 
travel 240 cm during each revo- 
lution, what force will be neces- 
sary to raise a bucket of coal 
weighing 40 k ? 

23. Through how many me- 
ters must the force act to raise 
the bucket from a cavity 10 m 

24. The truck (Fig. 80) is a lever ; the fulcrum is at the axle M of the 
wheels. AB represents the line of direction of the load, i.e. the direction 
in which the resistance acts ; and C D represents the direction in which 
a force acts to produce equilibrium in the load in its present position. 
a. What represents the force-arm ? b. What represents the resistance- 
arm ? c. The force required to support the load is what part of the 
load ? d. Would greater, or less, force be required if it were applied at 
E instead of C ? Why ? e. How may the load be supported without any 
force applied to the lever, the legs not touching the ground ? /. Would 
its equilibrium in this position be stable or unstable ? Why ? g. Suppose 
the feet F to rest upon the ground, how would the pressure of the load be 
distributed between the feet and wheels ? h. Which is better suited for 

Fig. 79. 



moving heavy burdens, a wheelbarrow or a truck ? Why ? i. Suppose 
that C D represents the supporting force and C G the force employed in 
moving the load, how would the intensity and direction of the single 
force that accomplishes both results be found ? 

25. A plank 12 feet long and weighing 24 pounds is supported by two 

Fig. 80. 

props, one 3 feet from one end, and the other 1 foot from the other end. 
What is the pressure on each prop ? 

26. What must be the diameter of a wheel in order that a force of 20 

Fig. 81. 

Fig. 82. 

pounds applied at its circumference may be in equilibrium with a resist- 
ance of 600 pounds applied to its axle, which is 3 inches in diameter ? 

27. How would you calculate the mechanical advantage gained by a 
machine like that of Fig. 81 ? (On the axle A is an endless screw a, by 
means of which motion is communicated from the axle to the wheel W.) 



28. a. Where is the fulcrum in a claw-hammer (Fig. 82) ? b. What 
is the ratio of the mechanical advantage gained by means of it ? 

29. In its technical meaning, a "perpetual motion machine " is not a 
machine that will run indefinitely, but a machine which can do work 
indefinitely without the expenditure of energy. Show that such a machine 
is impossible. 

Section XII. 


91. Comhination of pulleys. — As has been shown (p. 98), 
mechanical advantage is gained with the movable pulley. In 
Fig. 83, M, the cord passing around the movable pulley may 
be supposed to be divided into two parts each supporting half 


Fig. 83. 

Fig. 84. 

the load; the tension in each part is half the load, and the 
supporting force P is half the load. In Fig. 83, N, the movable 
block A contains two pulleys, the cord supporting the load is 
divided into four parts, and P is one fourth of L ; and generally 
with a combination of pulleys having a continuous cord 

i.e. the force is equal to the load divided by the number of parts 
of the cord supporting the movable block. 



92. Combination of levers. — In the arrangement shown in 
Fig. 84 there are three simple levers combined so as to form 
one coni])oiind lever. The supporting force is applied at A : 
the resistance applied to this simple lever at B is identical 
with the force applied at A', and so on. Now 

Continuous product of resistance-arms 
f — I \^ ±- ii . 

Continuous product of force-arms 

A combination of levers similar to this may be seen in 
scales used for weighing very heavy bodies such as the 
so-called platform "hay scales," in which a, comparatively 
small weight counterbalances the heavy load. 

93. Combinations of the 
ivheel and axle. — Fig. 85 
represents a train of wheels 
in gear. A train of wheels 
being analogous to a com- 
pound lever, the mechanical 
advantage gained is obvi- 
ously the ratio of the con- 
tinued product of the radii 
of the wheels to the con- 
tinued product of the radii 
Fig. 85. of the axles. 


1. When there are four movable pulleys in one block, what length of 
cord passes through the hands in raising a weight 6 inches ? 

2. What force must a man weighing 180 pounds use, to support him- 
self by means of the pulleys in Eig. 83, N ? 

3. The circumference of the circle described by the end of the lever of 
a screw is 9 feet, and there are three threads to the inch ; what pressure 
will a force of 100 pounds exert ? 

4. Suppose the lengths of the arms of the several levers in Fig. 84 
bear the following relations to each other: 5:1,4:1, and 3:1; what force 
applied at A will support 180 pounds at W ? 



5. a. Of how many and what simple machines does the crane (Fig. 86) 
consist ? 6, How would you calculate the mechanical advantage gained 
by it? 

6. a. In what sense are machines ' ' labor-saving ' ' ? 
is no machine labor-saving ? 

6. In what sense 





94. Gravitation is universal. — An unsupported body falls 
to tlie earth. This is evidence of an action or stress between 
the earth and the body. It has been ascertained by careful 
observation that when a ball is suspended by a long string by 
the side of a mountain, the string is deflected from the vertical 
toward the mountain in consequence of an attraction between 
the mountain and the ball. Delicate experiments show that 
very small bodies tend to approach one another, and that the 
tendency of a body to fall to the earth is but a single instance 
of a tendency existing in all kinds and quantities of matter. 

That there is an attraction between the sun and the earth, 
and the earth and the moon, is shown by their curvilinear 
motions. Tides and tidal currents on the earth are due to the 
attraction of the sun and the moon. This attraction which 
exists between all masses is called gravitation. When bodies 
under its influence tend to approach one another, they are said 
to gravitate. Since this attraction ever exists between all 
bodies, at all distances, it is called universal gravitation. The 
theory of universal gravitation was established by Newton. 
No concept in science rests on a surer foundation. 

95. Weight. — The attraction between the earth and a ter- 
restrial body is called the weight of that body. The weight 
of a body, therefore, is the measure of the force of attraction 
between the body and the earth. 

We do not know the cause of gravitation. Whether the 
seat of the attraction or energy is, as the language in common 
use indicates, in the bodies themselves, or whether it exists in 


some medium which we may suppose to surround all bodies 
and fill all intervening space, we do not know.^ 

96. Law of gravitation. — Methods first given by Newton 
in the Princijna, but too elaborate for our purpose, have 
established the fact that the magnitude of the attraction 
between any two bodies depends upon two things, their 
masses and the distance hetiveen their centroids. The Law of 
Universal Gravitation is as follows : — 

The attraction between every two bodies of matter in the 
universe vaines directly as the 'product of their masses, and 
inversely as the square of the distance between their centroids? 

Eepresenting the masses of two bodies by m and m\ the 
distance between their centers of mass by d, and the attrac- 
tion by g, this relation is expressed mathematically thus : 

g oc (varies as) —j^- For example, if the mass of either 

body be doubled, the product {inm'') of the masses is doubled, 
and consequently the attraction is doubled. If the distance 

^2 4/ 

between their centers of mass be doubled, then ( — 

the attraction becomes one-fourth as great. 

97. Galileo^ s exjoeriment. — Galileo let fall from the top of 
the leaning tower at Pisa iron balls of different masses, and 
found that they fell with equal acceleration and reached the 
ground at the same instant. This celebrated experiment 
established two important facts : — 

(1) At any given place the acceleration due to gravitation is 
independent of the mass of the falling body ; in other words, 

1 " It may turn out to be a property inherent in matter, and an exception to every 
known case ; but it is more probable that it is not an inherent property in matter at 
all, but a property due to a strain in the medium in which all matter is immersed." 
(Lodge.) The gravitation stress existing in this medium may be, in some respects, 
analogous to the stress which exists in a stretched rubber band, which would tend to 
bring together any bodies between which it might be stretched. 

2 The attraction between two gram-masses whose centroids are one centimeter 
apart is one fifteen millionth of a dyne. 


for all bodies at the same place the acceleration due to the 
earth's attraction is the same. 

Let / and f be the intensities of two attractive forces 
acting to move two bodies, whose masses are respect- 
ively m and m\ to the earth ; then (p. 40) 

f=m a, and f = m a^ 
but, as proved by Galileo's experiment, 
a = a, 

hence (diviam^) ^ = -^ , 

^ ^ f m 

and, in general, f^m, 

i.e. (2) the intensity of the earth^s attraction at the 

same place varies as the mass. 

In other words, the deductions from this experi- 
FiG. 87. j^rigj^^ r^pe . ^^ ^jia^i^ 2i\\ free bodies, whatever their 
mass, fall toward the earth with equal accelerations, and 
(2) that if one body possess twice the mass of another, twice 
the force is required to give it the same acceleration. 

Proposition (1) is seemingly contradicted by every-day 
experience, for if a coin and a piece of tissue paper be 
dropped from a hight they fall with very different velocities 
and accelerations. But if a coin and several bits of paper 
be placed in a long glass tube (Fig. 87), the air exhausted, 
and the tube turned end for end, it will be found that the 
coin and the papers fall in the vacuum with equal velocities. 
It is evident, then, that when there is a difference in the 
acceleration of falling bodies at the same place it is not due 
to the force of gravitation but to some other force, e.g. the 
resistance of the air. 

98. Variation of gravitation^ or g, on the earth'' s surface. — 
A spherical body of uniform density acts upon a particle out- 
side it as if the entire mass were collected at its center. If 
the earth were a homogeneous sphere and at rest, then the 


value of g would be constant at its surface since every point 
in it would be equidistant from the center. But the earth 
is a spheroid, its polar diameter being about 43 kilometers 
(nearly 27 miles) less than the equatorial diameter. Conse- 
quently the value of g is less in the equatorial than in the 
polar regions, i.e. a given body stretches a spring balance less 
as it is carried from either pole toward the equator. 1[?he loss 
of weight of any body due to this increase of distance from 
the center of the earth in being transported from the poles to 
the equator, is estimated to be ^\^ of its weight at the poles. 
But we have previously seen (p. 74) that the centrifugal 
force at the equator diminishes the weight of a body -^\^. 
Now in consequence of difference in distance from the center 
of mass of the earth and difference in velocity due to the earth's 
rotation, a body weighs at the equator s^g + 2^9 = \\^ less 
than at the poles. 

99. Weight above the earWs surface. — We infer from the 
law of gravitation that a body weighs more at the earth's 
surface than above it ; in other words, bodies become lighter 
as they are raised above the earth's surface. But since the 
force diminishes as the square of the distance from the center 
(not from the surface) of the earth, and as the surface is about 
4,000 miles from the center, the diminution for a few miles or 
for any distance which we are able to raise bodies is scarcely 
perceptible ; hence in all commercial transactions we may, 
without important error, buy and sell as if the weighing 
always took place at the same distance from the center of 
the earth, in which case mass is strictly proportional to 

100. Weight helow the eartKs surface. 

It may be demonstrated geometrically (Wood's Elementary 
Mechanics, p. 38, Barker's Physics, p. 117) that if a particle be 
placed anywhere within a homogeneous hollow spherical shell of 
matter, as at a, c, or d (Fig. 88), it is in a state of equilibrium in 


regard to the attraction of the matter of the enveloping shell, i.e. a 
body thus placed within a hollow shell of the earth, of whatever 
thickness, would be under the influence of balanced forces and 
hence weightless. Hence if a body be taken below the surface of 
the earth, as from a to a' (Fig. 89), it is practically placed within a 
hollow spherical shell of the earth, and therefore is freed virtually 

Fig. Sd. 

from the gravitation influence of this shell. Its weight is now 
wholly determined by the gravitation of the smaller mass m n. So 
that as a body is carried below the surface of the earth, it loses 
in weight as much as it would if it were being transferred to 
smaller and smaller earths ; consequently at the earth's center 
matter is under the influence of balanced forces and hence weight- 
less. If the earth were a perfect sphere and homogeneous, the 
weight of a body would diminish uniformly as the distance from 
the surface increased. 


1. a. Which is independent of mass, weight or acceleration ? 6. Which 
varies as the mass '? 

2. Why does a hundred-pound iron ball fall with no greater accelera- 
tion than a one-pound ball of the same material ? 

3. a. Which falls with greater acceleration in the air, an iron ball or 
a wax ball ? Why ? h. How would their accelerations compare in a 
vacuum ? c. Is acceleration independent of kind of matter ? 

4. If the earth's mass were doubled without any change of volume, 
how would it affect your weight ? 

5. On what principle may you determine that the mass of one body is 
ten times the mass of another body ? 

6. How many times must you increase the distance between the centers 
of two bodies that their attraction may become one-fourth as great ? 



7. If a body on the surface of the earth be 4,000 miles from the cen- 
troid of the earth, and weigh at this place 100 pounds, what would the 
same body weigh if it were taken 4,000 miles above the earth's surface ? 

8. The masses of the planets Mercury, Venus, Earth, and Mars are 
respectively very nearly as 7, 79, 100, and 12 ; assuming that the distance 
between the centers of the first two is the same as the distance between 
the centers of the last two, how would the attraction between the first 
two compare with the attraction between the last two ? 

9. What would be the answer to the last question 
if the distance between the centers of the first two 
were four times the distance between the centers of 
the last two ? 

10. Would the weight of a soldier's knapsack be 
sensibly less if it were carried on the top of his rifle ? 

11. If you hold a body on a spring-balance in an 
elevator, what effect will be noticed as you start to 
ascend ? What effect, as you start to descend ? Ex- 
plain. ' 

12. Let E (Fig. 90) represent the earth as a perfect 
homogeneous sphere of a radius of 4,000 miles, a. If 
a body at a weigh 1 pound, what would it weigh at ?n, 
1,000 miles below its surface ? h. What at ?2, o, and 
c, respectively 2,000, 3,000, and 4,000 miles below the 
surface ? c. What at 6, d, e, and i, respectively 4,000, 
8,000, 12,000, and 2,000 miles above the earth's sur- 
face ? 

13. a. What is a vertical line ? 6. What angle does it make with the 
frame of a spirit level when in position ? 

14. A body weighs 100 pounds at the earth's surface. At what two 
places would its weight on a spring balance be 50 pounds ? 

15. If the acceleration at sea-level be 32.2 feet, what is it 5 miles above 
sea-level ? 

Fig. 90. 




Section I. 


101. Minuteness of idarticles of matter. The molecule. — 
Physiology teaches us that, in order to smell any substance, 
we must take into our nostrils, as we breathe, small particles 
of that substance which are floating in the air. The air, for 
several meters around, is sometimes filled with fragrance from 
a rose. You cannot see anything in the air, but it is, never- 
theless, filled with a very fine dust that floats away from the 
rose. At sea the odor of rosemary renders the shores of 
Spain distinguishable long before they are in sight. A grain 
of musk will scent a room for many years, by constantly 
sending forth into the air a dust of musk. Though the 
number of particles that escape must be countless, yet they 
are so small that the original grain does not lose perceptibly 
in weight. 

These instances, and numerous others of common obser- 
vation, give us only a feeble conception of the minuteness 
of particles of matter and of its extreme divisibility. Yet 
the smallest particle of dust of rosemary or musk, and the 
smallest particle of any substance which can be obtained by 
any mechanical means, is very large in comparison with 
bodies called molecules, which, of course, are too small to be 
seen, but of whose existence we have ample evidence. A 
single simple example of the proofs of their existence, though 
by no means the most conclusive, must suffice at this place. 


Matter, e.g. gold or water, is either continuous as it appears 
to the eye or it is discontinuous, granular, composed of distinct 

particles (called molecules) somewhat as rep- ^ ^ 

resented in Fig. 91. Matter is compressible !-':'.-*'' .*••'•'. 

and expansible. On the supposition that ^'"rr^TE" ■.'•''••■"■• ■.••*• 
matter is continuous, these properties are •.'••'.•;••••.•'•'. 

unexplainable ; but on the supposition that " ':..•..'. v.-.-.;' "' 

matter is molecular, these properties are '^°''Ita?I" ;};•;•'; •;".U-:; 
easily explainable. A change of volume by •••■•■•■•■■■••••■.■■. 

contraction or expansion means simply a 
coming together or a separation of the molecules composing 
the body, as represented in Fig. 91. 

102. Theory of the constitution of matter. — For reasons 
which will appear as our knowledge of matter is extended, 
physicists have generally adopted the following theory of the 
constitution of matter : Every body of matter except the mole- 
cule is composed of exceedingly small disconnected particles, 
called molecules. No two molecules of matter in the universe 
are in permanent contact with each other. Every molecule is in 
quivering motion, Tnoving back and forth between its neighbors, 
hitting and rebounding from them. When we heat a body we 
simply cause the molecules to move more rapidly through their 
spaces; so they strike harder blows on their neighbors, and 
usually pjush them away a very little ; hence the body expands. 

103. Porosity. — If the molecules of a body are never in 
contact except at the instants of collision, it follows that 
there are spaces between them. These spaces are called ijores. 

All matter is porous ; thus water may be forced through the 
pores of cast iron ; and gold, one of the densest of substances, 
absorbs liquid mercury. 

Impenetrability^ may be affirmed of molecules, but not 

iThe doctrine of impenetrability declares that " Two bodies of matter cannot 
occupy tbe same space at the same time." In its strict scientific sense this doctrine 
is as axiomatic as the statement that " A body cannot be in two places at the same 


necessarily of masses. The term pore, in physics, is restricted 
to the invisible spaces that separate molecules. The cavities 
that may be seen in a sponge are not pores, but holes ; they 
are no more entitled to be called pores, than the cells of a 
honeycomb or the rooms of a house are entitled to be called, 
respectively, the pores of the honeycomb or of the house. 

By means of delicate calculations, physicists ascertain approxi- 
mately the probable size of the molecule. Lord Kelvin estimates 
that the diameter of the molecules of a gas cannot be less than one 
five-hundred-millionth of a centimeter. The minimum particle 
visible to the eye is a cube one four-thousandth of a millimeter on a 
side. Such a cube contains from sixty to one hundred million 

"The kinetic theory of gases (p. 271) teaches that in a cubic inch 

of any gas at atmospheric pressure and at ordinary temperatures 

there are about 3 X lO^o detached particles absolutely similar and 

' equal to one another. Here v^e reach the limit of our present 

knowledge as to division of matter." (Tait.) 

104. Atomic theory of imatter. Atoms. 

The theory given above assumes that the molecule is tlie limiting 
particle of possible physical division, i.e. the smallest particle of any 
substance which can preserve the properties of that substance ; 
hence the molecule is sometimes termed the "physicist's unit." 
The chemist finds it necessary to assume that the molecule is capable 
of a still further subdivision, a division which results in a complete 
change in the character of the substance operated on. Thus a 
molecule of sugar when subjected to chemical processes, which are 
virtually chemical divisions, yields carbon, hydrogen, and oxygen, 
substances entirely unlike sugar. These still smaller particles 
obtained by the division of the molecule are called atoms. The 
atomic theory assumes that the atom, as the word etymologically 
signifies, is indivisible, and it may be termed the " chemist's unit." 
An atom is indestructible and unchangeable. About seventy differ- 
ent kinds of atoms have been discovered. These constitute the 
so-called elementary substances. All other substances are compounds 
of certain of these elements of varying degrees of complexity. A 
molecule consists of a group of atoms bound together by chemical 


forces usually termed chemical affinity, and a mass consists of a 
group of molecules which may or may not be bound together by a 
physical or molecular force called cohesion. 

Section II. 


105. Solids, liqicids, and gases. — In popular language there 
are said to be three states of matter, the solid, the liquid, and 
the gaseous. 

Solids preserve a definite volume and shape when left to 
themselves ; liquids tend to preserve a definite volume only, 
while their shape conforms to that of the containing vessel ; 
gases tend to preserve neither a definite volume nor shape, but 
conform not only in shape but in volume to the containing 
vessel, no matter how large this may be. We may have a 
vessel half full of liquid, but a mass of gas always occupies 
the whole of a containing vessel, however small the quantity 
of gas. Gases tend to expand indefinitely and to assume an 
infinite volume with a correspondingly small density. Solids 
and liquids may have free surfaces, gases cannot retain per- 
manently a free bounding surface independent of the contain- 
ing vesseL 

Which of the three states any portion of matter assumes depends 
upon its temperature and pressure. Just as at ordinary pressures of 
the atmosphere water is a solid {i.e. ice), a liquid, or a gas {i.e. steam), 
according to its temperature, so any substance may be made to 
assume any one of these forms unless a change of temperature 
causes a chemical change, i. e. causes it to break up into other sub- 
stances. Eor example, wood cannot be melted, because it breaks 
up into charcoal, steam, etc., before the melting-point is reached. 
In order that matter may exist in a liquid (and sometimes in a solid) 
state, a certain definite pressure is required. Ice vaporizes, but 
does not melt {i. e. liquefy) in a space from which the air (and con- 
sequently atmospheric pressure) has been removed. Solid carbonic 


acid vaporizes, but does not melt unless the pressure is greater 
than the ordinary atmospheric pressure. Charcoal has been vapor- 
ized, but has never been liquefied, undoubtedly because sufficient 
pressure has never been used. 

As regards the temperature and pressure at v^hich different sub- 
stances assume the different states, there is great diversity. Oxygen 
and nitrogen gases liquefy and solidify only at extremely low tem- 
peratures ; and then, only under great pressure. On the other 
hand, certain substances, as quartz and lime, are liquefied only 
by the most intense heat. 

106. Fluids. — The term implies the property of flowing. 
Since both liquids and gases possess this property in an 
eminent degree in consequence of great freedom of motion of 
their molecules around one another, they are both included 
under the common term fluid. Further on it will be seen that 
one of the chief distinctions between a solid and a fluid is 
that the former possesses rigidity, while the latter does not. 

107. Vapo7'ous state ; critical state. 

Closer study makes evident that the above classification is purely 
r.rbitrary. The three states of matter sometimes merge into one 
another so that there remains no distinct line of demarkation 
betvt^een them. For example there is a state which may be regarded 
as intermediate between the liquid and the gaseous, called the 
vaporous state. If a substance in the gaseous form be compressed 
or cooled to such an extent that it will suffer but little further 
compression or cooling without passing into the liquid state, it 
possesses in this state certain peculiar properties and is known by 
the name of vapor. A vapor may be defined as a gas near its con- 
densing point. "When matter is at a temperature and under a pres- 
sure such that if heated a little more it becomes a vapor, or if 
allowed to cool a little more it becomes a liquid, it is said to be in 
the critical state. In this semi-liquid state the gaseous and liquid 
states meet and are indistinguishable. The highest temperature 
at which this occurs is called the critical temperature, and the 
highest pressure the critical pressure. A vapor may be defined, also, 
as any gaseous substance at a temperature below its critical temper- 


For example, carbon dioxide at 31° and under a pressure of 73 
atmospheres is in a critical state. Heated a little it certainly becomes 
gaseous ; cooled a little it as certainly assumes the properties of a 
liquid, since it is far less compressible. But if the pressure be 
maintained, the transition from one to the other is not recognizable. 

108. Ultixigaseous or fourth state of matter. 

Air has been rarefied to the three-hundred-millionth of its normal 
density. But when gaseous matter is rarefied to even a millionth 
of the density of air at sea-level, it exhibits extraordinary properties 
quite as different from the gaseous as this is from the liquid state, 
so that some are disposed to consider that there is an ultragaseous 
or fourth state of matter. 

Section III. 


109. Molecular attractive forces. — Many of the properties 
of matter are due to molecular forces, some of wliicli now 
demand our attention. For convenience we call bodies of 
appreciable size molar (massive) bodies, or masses, in dis- 
tinction from molecules (bodies of very small mass). Action 
between molar bodies, usually at sensible distances, is called 
molar force ; action between molecules, always at insensible 
distances, is called m^olecular force. According to the theory 
of the constitution of matter the molecules of every mass are 
in ceaseless motion, hitting and rebounding from one another. 
This tends to drive the molecules apart. In gaseous masses 
the molecules move without restraint ; hence gaseous bodies 
always tend to expand. 

In solids and liquids the molecules are held under the 
action of a very powerful attractive force, called cohesion, 
which prevents their separation except under the action of 
considerable external force. It is the force which resists an 
effort tending to break, tear, or crush a body. The tenacity or 


tensile strength of solids and liquids, i.e. the resistance which 
they offer to being pulled apart, is due to this force. It is 
usually greater in solids than in liquids, and is entirely 
wanting in a true gas. 

110. Strain, rigidity, elasticity. — Strain means change of 
size, change of shape, or deformation of any kind. Change- 
of-size strain is called compression or dilatation, and the 
resistance of matter to it is called elasticity of volume. 
Change-of-shape strain, such as in flexion, torsion, etc., is 
called distortion, and the resistance to it manifests itself 
either as elasticity of figure, or rigidity. 

Elasticity is that property in vii'tue of ivhich a solid tends to 
recover its size and shape, and a fluid its size, after defor- 
mation. Solids are remarkable for high rigidity. A per- 
fectly rigid solid is one which, when a force is applied to it 
in any way, suffers no strain before breaking. No body is 
absolutely rigid, though some bodies are approximately so. 
If the stress between the molecules in opposition to the 
distorting force continue constant, regardless of the time the 
strain is kept up, and restore the body to its normal con- 
dition immediately on the removal of the distorting force, 
without any permanent strain or "set," the body is said to 
be perfectly elastic. All fluids are perfectly elastic, and a 
few solids are approximately so, such as ivory, steel, and 

If a solid have little or no tendency to recover its size and 
shape after distortion, it is said to be plastic or inelastic.^ 
Such substances are putty, wet clay, and dough. A great 
number of substances are elastic when the distorting forces 
are small, but break or receive a " set " when these forces 
are too great. They are said to be elastic '^ within certain 
limits," called the limits of elasticity. If strained beyond 
those limits, they become more or less plastic. Hence the 
springs of a buggy sometimes become set from bearing a too 


heavy load and lose permanently mucli of their elasticity ; 
i.e. they become in a degree plastic. 
111. Viscosity. 

Experi77ient 1. — Support in a horizontal position, by one of its extrem- 
ities, a stick of sealing-wax, and suspend from its free extremity an 
ounce weight, and let it remain in this condition several days, or perhaps 
weeks. At the end of the time the stick will be found permanently bent. 
Had an attempt been made to bend the stick quickly, it would have been 
found quite brittle. 

It may seem like an abuse of the term to call sealing-wax 
a fluid, yet the experiment shows it to be a fluid, or at least 
to possess fluidity, or freedom of motion of its molecules 
around one another, in a small degree. Eesistance to deforma- 
tion due to the friction of the molecules of a body in sliding 
over one another is called viscosity. Bodies that slowly suffer 
continuous and permanent deformation under the action of a 
continuous stress are said to be viscous. A lump of pitch in 
course of time loses its sharpness of outline and flows down 
hill of its own weight. It is very viscous. Cold molasses is 
quite viscous, but as its temperature is raised its viscosity 
diminishes and it becomes more and more plastic or mobile. 
A perfectly rigid solid is one of infinite viscosity. A jperfect 
fluid is a fluid which possesses no viscosity. Gases are 
viscous to some extent and are therefore imperfect fluids. 

Bodies surrounded by air have on their surfaces an adherent film 
. of air. When they move, this film rubs against the surrounding air, 
and thus their movements are retarded by friction in the air. To 
the viscosity of the air is due in part the retardation of the velocity 
of falling bodies. A penny and a piece of tissue paper fall with 
equal accelerations in a vacuum (p. 120), but in the air the penny 
falls more rapidly because it presents less surface and therefore is 
retarded less in proportion to its weight by the friction of the air. 

Falling water is retarded by the air ; conversely, air is dragged 
down by falling water, as shown by the following experiment. 






Experiment 2. — Take a long glass tube A (Fig, 92), funnel-shaped 
at one end, and having a little below this end 
a short branch tube a. Pour water into the 
funnel and observe that it falls in a con- 
tinuous stream until it reaches a, but below 
this point it is broken up by descending 
bubbles of air which it drags along with it 
after coming in contact with the air at a. 
K direct sunlight be allowed to strike the 
tube, the interior reflection from the water 
in this part of the tube is dazzling and 

Connect with a by means of a rubber 
tube another short piece of glass tube 6, and 
introduce the latter into a tumbler of water. 
The water descending in A exhausts the air 
in 6 and the outside atmospheric pressure 
causes the water in the tumbler to ascend 
this tube 6, and thus the tumbler may be 
emptied. If a closed vessel B be connected with a, a high vacuum 
may be obtained in it. 

Fig. 92. 

Experiment 3. — Pour water from an elevation upon a still body of 
water below and observe the bubbles of air which form in the water, the 
air being dragged by the falling current to a considerable depth in the 

112. Hardness. — Hardness is resistance to abrasion or 

To enable ns to express degrees of hardness, the following 
table of reference is generally adopted : — 


1. Talc. 

2. Gypsum (or Rock-Salt). 

3. Calcite. 

4. riuor-Spar. 

5. Apatite. 


Orthoclase (Feldspar). 






By comparing a given substance with the substances in the 
table, its degree of hardness can be indicated approximately. 
Thus '-11 = 7" means that the body is about as hard as quartz. 

113. Malleability, ductility. — Solids which possess that 
kind of fluidity which renders them susceptible of being rolled 
or hammered out into sheets are said to be malleable. Most 
metals are highly malleable. Gold may be hammered so thin 
as to be transparent, or to a thickness of one three-hundred- 
thousandth of an inch. Most substances that are malleable 
are susceptible also of being drawn out into fine threads, 
e.g. wire of different metals. Such substances are said to be 
ductile. Platinum has been drawn into wire .000165 inch 
thick, or so fine as to be scarcely visible to the unaided eye. 

Section IV. 


114. Cohesion of liquids. — Clean glass is wet by water. 
If a glass plate be dipped into water and then withdrawn, a 
layer of water clings to the glass. When the glass is with- 
drawn, water is torn from water, and not glass from water. 
This shows that the attraction of the molecules of water for 
one another is weaker than the attraction between glass and 
water. Or if, to save words, we call the attraction between 
the solid and the liquid adhesion, then we may say that the 
cohesion between the molecules of the water is weaker than 
the adhesion between the glass and the water. 

Clean glass is not wet by clean mercury, which shows that 
the adhesion between glass and mercury is not as great (about 
one third as great) as the cohesion in mercury. Generally 
speaking a solid is wet by a liquid when the adhesion of the 
solid to the liquid is greater than the cohesion of the liquid, 
and is not wet when the cohesion is greater than the adhesion. 


115. Surface tension. — When a rubber band is strained or 
stretched, it is said to be in a state of tension, and there exists 
between its molecules a contractile or resilient stress. 

In liquids the molecules are within the limits of one another's 
attractions, which accounts for a greater or less viscosity or 
hindrance of flow and also for a certain phenomenon called 
surface tension. Every liquid behaves as if a thin film forming 
its external layer were in a state of tension, or were exerting 
a constant effort to contract. 

It is not within the scope of this book^ to explain in full 
the dynamics of the molecular forces by which this result is 
brought about ; it must suffice to call the attention of the 
student to the peculiar condition, with reference to mutual 
attractions, of those molecules which compose the surface film. 
In the interior of a liquid each molecule is surrounded by 
other similar molecules and the position which it assumes is 
that in which it is acted upon equally in all directions, and 
there is nothing to render the mutual attractions manifest. 
At a free surface, however, the molecules can be acted upon 
only by others lying internal to them. The result is a system 
of forces acting at right angles to the free surface of the 
liquid, and tending to reduce the free surface to the least 
possible area. This tendency of a liquid surface to contraction 
means that it acts like an elastic membrane, equally stretched 
in all directions, and by a constant tension. In the case of 
pure water at 20° C. this tension is about 81 dynes per linear 

Experiment. — Form a soap-bubble at the orifice of the bowl of a 
tobacco pipe, and then, removing the mouth from the pipe, observe that 
the tension of the two surfaces (exterior and interior) of the bubble drives 
out the air from the interior and finally the bubble contracts to a flat 
sheet of minimum area. 

^ The student who is desirous of knowing more of this interesting subject may 
consult the article " Capillarity," by Maxwell, in the Encyclopsedia Britannica. 


As a consequence of surface tension, eyery body of liquid tends to 
assume the spherical form, since the sphere has less surface than any- 
other form having equal volume. In large bodies the distorting forces 
due to gravity are generally sufficient to disguise the effect ; but in small 
bodies, as in drops of water or mercury, it is apparent. Again, if the 
distorting effect of weight be eliminated in any way, as by immersing a 
quantity of oil in a mixture of water and alcohol of its ovrai density, or by 
replacing the central portion of the body by a fluid much lighter than its 
own kind, as in the case of a soap-bubble, the sphere is the resulting form. 

116. Cajjillari/ 2yhenonie7ia. — Surface tension is by no means 
peculiar to liquids. The surfaces of all bodies tend to con- 
tract. But since gases have no surfaces of their own, and the 
rigidity of solids prevents an alteration of shape, it is obvious 
why liquids show the effects of surface tension most readily. 
But the surface tensions of solids and gases perform their 
part in determining certain phenomena. For example, if a 
glass rod be thrust vertically into water so as to leave a jjart 
projecting into the air, the surface of the water does not meet 
the rod at right-angles, but is turned up so as to form a very 
small angle ^ with the surface of the glass, as acb (Fig. 93). 
Here the three substances, water, glass, and air, are brought 
in contact and there are a triplet of tensions in operation the 
resultant of which is a force which pulls the water up against 
the glass wall. On the other hand if mercury, glass, and air 
be brought in contact, the relation between the triplet of 
forces becomes so changed as to cause the mercury to meet the 
glass at a- very large angle, about 135°. It thus seems that 
when a solid, a liquid, and a gas are in contact, their boundary 
surfaces form contact angles with one another determined by 
their relative surface tensions. 

If a glass tube x (Fig. 94) of capillary (hair-like) bore be 
thrust into water, the water will rise in the bore considerably 
above the general level outside. If a similar tube y (Fig. 95) 

1 If the glass be quite clean tlie angle is 0. If not clean, it may reach, and even 
exceed, 90°. • 



be thrust into mercury, the mercury within the bore will be 
depressed below the surface outside. Phenomena of this kind 
are called capillary idhenomena. The surfaces of the liquids 
inside the bores are curved, the surface of water being concave 
and that of mercury convex. The size of the bore of the 

Fig. 94. 


a A i 1 e 

Fig. 96. 

tubes X and y is greatly exaggerated in order to show this 
more plainly. The concavity and convexity of these interior 
surfaces are a necessary consequence of the angles of contact 
with which these liquids meet glass. It remains only to 
explain the elevation and depression of the column of liquid 
in the tube. This may be done in part by analogy. Let AB 
(Fig. 96) represent a clothes line suspended slackly between 
two posts. From this line hang by strings small stones 
a, h^ c, etc. If the hempen line become wet, as in a rain, it 



contracts and straightens, as shown by the dotted line AB. 
In other words, the contractile force which is exerted obliquely 
{e.g. nm Fig. 96) is resolvable into two forces, one of which 
is horizontal and the other is vertically upward ; the latter 
tends to elevate the stones. In a similar manner the curved 
surfaces of water and mercury tend to contract and become 
flat. In the case of the water surface (which is concave) the 
contractile force tends to elevate the pendent liquid ; but in 
the case of the mercury surface (which is convex) the tendency 
is to produce depression. On the nature of the curvature 
depends the direction in which the contractile force acts on 
the pendent liquid. Now it is evident that water will be 
drawn up by this contractile force until the weight of the 
column balances this force ; and mercury will be depressed 
until the force is balanced by the pressure of the mercury out- 
side the tube. Capillary phenomena are, therefore, phenomena 
of surface tension. 

The phenomena of capillary action are well shown by placing 
various liquids in U-shaped glass tubes having one arm 
reduced to a capillary size, as A 
and B in Fig. 97. Mercury poured 
into A assumes convex surfaces 
in both arms, but does not rise as 
high in the small arm as it stands 
in the large arm. Pour water into 
B, and all the phenomena are re- 
versed. Fig. 98 shows the forms 
that the surfaces of water and 
mercury take when contained in the same glass tube. 

The following laws of capillary action may be verified by 
experiment : — 

I. Liquids rise in tubes tvhen they wet them, and are de^jressed 
when they do not. 


Fig. 97. 



II. The elevation or depression varies inversely as the diameter 
of the bore. 

III. The elevation and depression vary with the nature of the 
liquids emjjloyed, and with the substance of the tube. 

lY. The elevation or depression varies inversely ivith the tem- 

Section V. 


117. Other molecular phenomena. 

Besides the phenomena we have just studied, there are a great 
many others depending in part on molecular attraction, but much 
more on molecular motions, of which we learned on p. 125 and 
which we now must consider more in detail. 

The molecules of all bodies are constantly in a state of motion. 
The higher the temperature the greater are their velocities. In a 
solid "a molecule, though in continual motion, never gets beyond 
a certain very small distance from its original position in the body. 
In fluids there is no restriction to the excursions of a molecule. 
True the molecule travels only a very small distance before it 
encounters another molecule ; but after this encounter there is 
nothing which determines the molecule rather to return towards the 
place whence it came than to push its way into new regions. Hence 
in fluids the path of a molecule is not confined to a limited region, 
but may penetrate to any part of the space occupied by the fluid." 

118. Diffusion of liquids. 

Experiment 1. — Partially fill a glass jar (Fig. 99) with water. 
Then introduce beneath the water, by means of a long tunnel, a 
concentrated solution of sulphate of copper. The lighter liquid 
rests upon the heavier, and the line of separation between the two 
liquids is at first distinctly marked. But in the course of days or 
weeks this line will gradually become obliterated, the heavier blue 
liquid will gradually rise, and the lighter colorless liquid will 
descend, till they become thoroughly mixed. 

Experiment 2. — Take about 1 cc of bisulphide of carbon, color 
it by dropping into it a small particle of iodine, and pour this 



colored solution into a test-tube nearly filled with water. The 

colored liquid, being heavier than the water, sinks 

directly to the bottom, and . shows no tendency to 

mix with the water. But in the course of time you 

discover that the colored liquid diminishes in quantity, 

and finally disappears. The peculiar odor of this 

substance which pervades the air in the vicinity shows 

that a considerable portion has evaporated. But it 

must have worked its way gradually through the 

water above it. 

If, in the last two experiments, you examine the 
liquid with a microscope during the operation, you 
will not be able to trace any currents ; hence the motion of the liquids 
is not m mass, but by molecules — a true intermolecular motion. 

An intermingling of the molecules of two liquids caused by their 
own motions is called diffusion of liquids. 

119. Diffusion of gases. 

Experiment 3. — Fill a test-tube with oxygen gas, and thrust into 
it a lighted splinter ; the splinter burns much more rapidly than in 
the air. Fill another tube with hydrogen gas, and keep 
the tube inverted (for, this gas being about 14.4 times 
as light as air, there will be no danger of its falling out). 
Thrust in a lighted splinter ; the gas takes fire, and burns 
with a pale flame at the mouth of the tube. Next fill 
one tube with oxygen and the other with hydrogen 
gas, and place the mouth of the latter over the mouth 
of the former, as in Fig. 100. In about a minute apply 
a lighted splinter to the mouth of the tube (let the mouth 
of each tube be freely open to prevent accident) ; a slight 
explosion takes place in each instance. It is apparent 
that although the oxygen gas is 14.4 times as heavy as 
the hydrogen, some of it has risen into the upper tube, 
while some of the lighter , hydrogen has descended into 
the lower tube, and the two gases have become diffused. 

There are liquids between which diffusion does not 
take place. But it does take place between any two 
gases whenever they are placed in contact. 

In consequence of this universal tendency to diffusion, 

gases will not remain separated — i.e. a lighter resting 

Fig. 100. upon a heavier, as oil rests upon water. This is of 



immense importance in the economy of nature. The largest portion 
of our atmosphere consists of a mixture of oxygen and nitrogen 
gases. There are always present also small quantities of other gases, 
such as carbon dioxide, ammonia gas, and various other gases, 
which are generated by the decomposition of organic matter. These 
gases, obedient to gravity alone, would arrange themselves according 
to their weight, — carbonic-acid gas at the bottom, or next the earth, 
followed respectively by oxygen, nitrogen, ammonia, and other gases. 
Neither animal nor vegetable life could exist in this state of things. 
But, in consequence of the diffusibility of these gases, they are 
found intimately mixed and in the same relative proportions, 
whether in the valley or on the highest mountain peak. 

120. Osmose, or diffusion of liquids through membranous septa. 

Certain liquids when separated from each other by membranous 
septa diffuse through these septa, and the diffusion may be more 
rapid than when no septa intervene. Such membranes as a bladder, 
cow's pericardium, and parchment paper are especially suited to 
this purpose. Diffusion through septa is called osmose. There is 
a wide range of relative diffusibilities. At one end of the scale are 
such substances as solutions of urea, common salt, and such sub- 
stances as crystallize, hence called crystalloids. These diffuse 
^ rapidly. At the other extreme are such substances as 
starch, gum, albumen, gelatine, and glue-like matter, 
called colloids. These diffuse very slowly. If a mixture 
composed of a crystalloid and a colloid be placed in a 
bladder, for instance, and the bladder be suspended in 
water, the crystalloid will rapidly diffuse through the 
septum while the colloid will diffuse very slowly. In 
this way a separation of the two substances may be 
eft'ected. Separation by this process is called dyalysis. 

121. Osmose of gases. 

Experiment 4. — Seal the open end of a thin, unglazed 

earthen cup, such as is used in a Bunsen battery (p. 

471), with plaster of Paris, through which extends a 

glass tube. Place the exposed end of the tube in a 

Fig. 101. cup of colored water. Lower a glass jar filled with 

hydrogen or coal-gas over the porous cup, as in Fig. 101. Instantly 

air is forced down through the tube, and escapes in bubbles from 



the colored liquid. The gas in the larger vessel forces its way 
through the pores of the cup, diffuses through the air contained in 
it, and causes an unusual pressure on the colored liquid, as is 
evinced by the air that is forced out through it. In a minute remove 
the glass jar. The hydrogen now escapes through the sides of the 
cup, and mixes with the air on the outside ; a partial vacuum is 
formed in the cup, and water rises in the tube. In each case air 
passes through the sides of the porous cup, but the influx and efilux 
of the hydrogen is much more rapid than that of the air. 

An interesting modification of this apparatus is the diffusion 
fountain (Fig. 102). By passing the glass tube of the porous cup 
through the cork of a tightly-stoppered vessel, 
and having another glass tube pass through 
another perforation in the same cork, water is 
forced out in a jet several feet in hight, when 
the hydrogen jar is held over the porous cup. 

Children well understand that toy balloons, 
which are made of collodion and filled with 
coal-gas, collapse in a few hours after they are 
inflated. This is caused by the escape of the gas 
by osmose. Nature furnishes an illustration of 
osmose of gases in respiration. In the lungs 
the blood is separated from the air by the thin, 
membranous walls of the veins. Carbon dioxide 
escapes from the blood through these septa, and oxygen gas enters 
the blood through the same septa. 

The phenomena of diffusion both in liquids and gases furnish strong 
and tangible evidence that these bodies consist of molecules in a state 
of continual motion. 

Fig. 102. 





Section I. 


122. Laiu of hydrostatiG and 'pneumatic transmission of 
jjressure. — That branch of science which treats of liquids in a 
state of equilibrium or rest is called hydrostatics ; that branch 
which treats of liquids in motion is called hydrokinetics ; 
and that branch which treats of the dynamics of air and other 
gases is called pneumatics. With the 
exception of phenomena occasioned 
by difference in compressibility and 
expansibility, liquids and gases are 
subject to the same laws and may 
be treated togetlier, in so far as they 
are alike, under the common term 

Experiment 1. — Fill the glass globe and 
cylinder (Fig. 103) with water, and thrust 
the piston into the cylinder. Jets of water 
will be thrown not only from that aperture 
a in the globe toward which the piston 
moves and the pressure is exerted, but from . 
all the apertures. 

If a finger be placed loosely over 
the end of a water faucet, spray will 
be thrown to equal distances in all 
directions. It thus appears not only 
that external pressure is exerted 
upon that portion of the liquid that lies in the path of the 

Fig. 103. 


force, but that it is transmitted equally to all parts and in all 

When pressure is exerted upon a solid, on account of its 
rigidity it is incapable of transmitting the pressure in other 
than the direction in which it is pressed. With fluids it is 
widely different. On account of the mobility of their mole- 
cules, they are incapable of resisting a change of shape when 
acted upon by a force which is not equally applied over the 
whole surface of the body of fluid, hence any force applied 
to a fluid body must be transmitted by the fluid in every 
direction. Consequently every portion of the interior walls 
of the containing vessel with which the fluid is in contact is 
subjected to pressure. 

Experiment 2. — Measure the diameter of the bore of each arm of the 

glass U-tube (Fig. 104). We will suppose, for illustration, that the 

diameters are respectively 40 mm and 10 mm ; then the ratio of the areas 

of the transverse sections of the bores will be 40^ : 102=16 ; that is, when 

the tube contains a liquid, the area of the free surface of the liquid in the 

large arm will be 16 times as great as of that in the small arm. Pour 

mercury into the tube until it stands about 1 cm 

above the bottom of the large arm as cd. The 

mercury stands at the same level in both arms. 

Pour water upon the mercury in the large arm until 

this arm lacks only about 1 cm of being full, as a b. 

The pressure of the water causes the mercury to 

rise in the small arm, and to be depressed in the 

large arm. Pour water very slowly into the small 

arm from a beaker having a narrow lip, until the 

Fig 104 
surfaces of the water in the two arms are at the 

same level. It is evident that the quantity of water in the large arm is 

16 times as great as that in the small arm. 

This phenomenon appears paradoxical (cipparently contrary 
to the natural course of things), until we master the important 
hydrostatic principle involved. We must not regard the body 
of mercury as serving as a balance beam between the two 
bodies of water, for this would lead to the absurd conclusion 






^■r ), 



that a given mass of matter may balance another mass 16 
times as great. We may best understand this phenomenon 
by imagining the body of liquid in the large arm ^ to be 
divided into cylindrical columns of liquid of the same size as 
that in the small arm. There will evidently be 16 such 
columns. Then whatever pressure is exerted on the mercury 

Fig. 105. 

by the water in the small arm is transmitted by the mercury 
to each of the 16 columns, so that each column receives an 
upward pressure, or a supporting force equal to the weight of 
the water in the small arm. 

The pressure exerted by a fluid upon the vessel containing 
it is normal to the walls of the vessel. Fluid pressure is 
expressed by stating the force exerted on a unit area, as 2 lbs. 
per sq. in., 5 g per cm^, etc. The total pressure on any surface 



is the product of the pressure per unit area multiplied by the 
number of units of area. 

Experiment 3. — Fig. 105 represents a section of an apparatus called 
(from the number of uses to which it may be put) the seven-in-one appa- 
ratus. A is a hollow cylinder closed at one end. B is a tightly fitting 
piston which may be pushed into or drawn out of the cylinder by the 
handle C when screwed into the piston. D is another handle permanently 
connected with the closed end of the cylinder. E is a nipple, opening 
into the space below the piston. To this may be attached a thick- walled 
rubber tube F. G is a stop-cock and H is a funnel, either of which may 
be inserted at will into the free end of the tube. 

Support the seven-in-one apparatus with the open end upward, force 
the piston in, place on it a block of wood A (Fig. 106), and on the block 
a heavy weight. Attach one end of the rubber tube 
B (12 feet long) to the apparatus, and insert a funnel 
C in the other end of the tube. Raise the latter 
end as high as practicable, and pour water into 
the tube. Explain how the few ounces of water 
standing in the tube can exert a pressure of many 
pounds on the piston, and cause it to rise together 
with the burden that is on it. 

Fig. 106. 

Fig. 107. 

Experiment 4. — Remove the water from the apparatus, place on the 
piston a 16-pound weight, and blow (Fig. 107) from the lungs into the 
apparatus. Notwithstanding that the actual pushing force exerted 
through the tube by the lungs probably does not exceed a few ounces, 
the slight increase of pressure caused thereby, when exerted upon the 
(about) 26 square inches of surface of the piston, causes it to rise together 
with its burden. 



A pressure exerted on a fluid enclosed in a vessel is trans- 
mitted undiminished to every part of that vessel ; and the total 
pressure exerted on the interior of the vessel is equal to the area 
multiplied hy the pressure per unit of area. 

123. The hydraulic press. — Closely allied to the seven-in- 
one apparatus is the hydraulic press. It contains two pistons, 
t and s (Fig. 108). The area of the lower surface of t is (say) 
one hundred times that of the lower surface of s. As the 
piston s is raised and depressed, water is pumped up from 
the cistern A, is forced into the cylinder x, and exerts an 

upward pressure against the 
piston t one hundred times 
greater than the downward 
pressure exerted upon s. 
Thus, if a pressure of one 
hundred pounds is applied 
at s, the cotton bales will be 
subjected to a pressure of 
five tons. 

The pressure that may be 
exerted by these presses is 
enormous. The hand of a 
child can break a strong iron 
bar. But observe that, al- 
though the pressure exerted 
is very great, the upward movement of the piston t is very 
slow. In order that the piston t may rise 1 cm, the piston s 
must descend 100 cm. The disadvantage arising from slowness 
of operation is insignificant, however, when we consider the 
great advantage accruing from the fact that one man can 
produce as great a pressure with the press as a hundred men 
can exert without it. 

The press is used for compressing cotton, hay, etc., into 
bales, and for extracting oil from seeds. The modern engineer 

Fig. 108. 


finds it a most efficient machine whenever great resistances 
are to be moved through short distances. 

124. Fi^essure of fluid due to its iveight. — Fluids exert 
pressure due to their weight. Imagine a vessel filled with 
shot ; you will understand that the upper layer of shot will 
press upon the layer next beneath with a force equal to its 
weight, the second upon the third with a force equal to the 
sum of the weights of the first two, and so on. You will also 
readily conclude that the pressure exerted upon the successive 
layers will be exactly proijortional to their d&pths^ unless in 
consequence of the great pressure to which the lowest layers 
are subjected there should be a crowding together of the shot 
so as to make them more compact. In this case there would 
be a slight variation from the rule as stated. For a like reason 
the downward pressure in a body of liquid increases as its 
dejjth except in so far as the pressure is modified in conse- 
quence of the compressibility of liquids. Liquids are, how- 
ever, so slightly compressible that any variation in conse- 
quence of the compression is usually neglected, and the 
principle is stated in general that pressure at any point in a 
liquid varies as its depth. 

Since the shot possess a certain degree of mobility or free- 
dom of motion around one another, their weight will cause to 
some extent a lateral pressure against one another and against 
the walls of the con- __-=_ 

taining vessel. In 
consequence of the 

extreme mobility of . ^Q^^^6^=^-r^=^-f^^=^ 
the molecules of fluids 
the downward pres- 
sure due to gravita- 
tion at any point in fig. 109. 
a fluid gives rise to an equal pressure at that point in all 
directions. Hence the so-called PascaVs principle: At any 
point in a fluid at rest the pressure is equal in all directions. 



Thus, let a, h, c, etc. (Fig. 109), represent imaginary surfaces, 
and the arrow-heads the direction of pressure exerted at 
points in these surfaces at equal depths in a liquid. The 
pressures exerted at these several points are equal. 

The truth of this principle is obvious, for if there be any 
inequality of pressure at any point, the unbalanced force will 
cause particles at that point to move, which is contrary to the 
supposition that the fluid is at rest. Conversely, when there 
is motion in a body of fluid it is evidence of an inequality 
of pressure. 

125. Methods of calculating liquid pressure. — Conceive of 

a square prism of water (Fig. 110), in the midst of a 

body of water, its upper surface coinciding with the free 

.^^^ m^^ ^^=^^ ^^^ - surface of the liquid. Let 

^^g ^^^ ^ p^^^ ^ y" the prism be 4 cm deep and 

j^M " "' ^^y " 1 cm square at the end ; then 

the area of one of its ends is 
1 cm^ and the volume of the 
prism is 4 cc. Now the weight 
of 4 cc of water is 4 g, hence 
this prism must exert a 
downward pressure of 4 g 
upon an area of 1 cm^ But 
at the same depth the pres- 
sure in all directions is the 
same, hence, generally, the 
pressure at any depth in 
water may be taken approxi- 
mately as one gram per 
square centimeter for each 
centimeter of depth (=c= 1,000 k per m^ for each meter of 
depth; or, since the weight of water is about 62.3 lbs. per 
cu. ft., the pressure is 62.3 lbs. per square foot for each foot 
of depth). In any other liquid, to determine the pressure at 

Fig. 110. 


any depth the water pressure at the given depth must be 
multiplied by the specific density (p. 177) of the liquid. 

The conclusions arrived at may be summarized as follows : 
In a mass of liquid at rest, the pressure is the same at all ^points 
in any horizontal plane, and is equal to the weight of a column 
of the liquid one square centimeter in section extending vertically 
from the horizontal plane to a horizontal plane coinciding with 
the upper surface of the liquid. 

Experiment 5. — Take a square prism of pine wood 1 cm square at 
its ends. Find its weight in grams and calculate to what depth it would 
sink endwise in water. Take a test tube a little larger and longer than 
the prism. Nearly fill the tube with water, lower the prism into the 
water, and verify your calculation. How great is the upward pressure of 
the water upon the bottom of the prism ? 

Fig. 112. 

Fig. 113. 

Fig. 114. 

Experiment 6. — A and B (Fig. Ill) are two bottomless vessels which 
can be alternately screwed to a supporting ring C (Fig. 112). The ring 
is itself fastened by means of a clamp to the rim of a wooden waterpail. 


A circular disk of metal D is supported by a rod connected with one 
arm of the balance-beam E. When the weight F is applied to the other 
arm of the beam, the disk D is drawn up against the ring so as to supply 
a bottom for the vessel above. Take first the vessel A, screw it to the 
ring and apply the weight to the beam as in Fig.' 114. Pour water slowly 
into the vessel, moving the index a up the rod so as to keep it just at the 
surface of the. water, until the downward pressure of the water upon the 
bottom tilts the beam, and pushes the bottom down from the ring, and 
allows some of the water to fall mto the pail. Eemove vessel A, and 
attach B to the ring as in Fig. 113. Pour water as before into vessel B ; 
when the surface of the water reaches the index a, the bottom is forced 
off as before. 

That is, the pressures upon the bottoms of all vessels {of whatever 
capacity or shape) are the same, provided the bottoms be of the same area 
and the depth and density of the liquid be the same. 

126. Rules for calculating liquid j^ressure agaiiist the bottom, 
and sides of a contcmiing vessel. — The total 'pressure due to 
gravity on amj portion of the horizontal bottom of a vessel con- 
taining a liquid is equal to the iveight of a column of the same 
liquid ivhose base is the area of that portion of the botto^n pressed 
upon, and whose hight is the depth of the ivater in the vessel. 
Thus, suppose that we have three vessels having bottoms of the 
same size : one of them has flaring sides, like a wash-basin ; 
another has cylindrical sides ; and the third has conical sides, 
like a coffee-pot. If the three vessels be filled with water to 
the same depth, the total pressure upon the bottom of each 
will be equal to the weight of the water in the vessel of 
cylindrical shape. Suppose that the area of the bottom of 
each is 108 square inches, and the depth of water is 16 inches ; 
then the cubical contents of the water in the cylindrical vessel 
is 1,728 cubic inches, or 1 cubic foot. The weight of 1 cubic 
foot of water is 62.3 pounds. Hence, the total pressure upon 
the bottom of each vessel is 62.3 pounds. 

Evidently, the lateral pressure at any point of the side of 
a vessel depends upon the depth of that point ; and, as depth 
at different points of a side varies, to find the total pressure 


upon any portion of a side of a vessel, find the weight of a 
column of liquid whose hase is the area of that portion of the 
side, and luhose hight is the average depth of that portion, 

127. The surface of a liquid at rest is level. -—By jolting a 
vessel the surface of a liquid in it may be made to assume tlie 
form seen in Eig. 115. Can it retain this form ? Take two 
molecules of the liquid at the points a and h on the same level. 
The total downward pressures upon a and h are , ^ 

as their respective depths ca and dh. But l^^^^plj 
since (assuming the mass of liquid to be at rest) pi|ip|if||ji|||;li 
the pressure at a given depth is equal in all i^nHIPjIf!! 
directions, c a and d b represent the lateral pres- 1^— ^~»— ^i 
sures at the points a and b res]3ectively. But 
db is greater than ca ; hence, the molecules a and h, and those 
lying in a straight line between them, are acted upon by two 
unequal forces in opposite directions. Hence the liquid can- 
not be at rest in the position assumed and there will, therefore,, 
be a movement of molecules in the direction of the greater 
force, toward a, till there is equilibrium of forces, which will 
occur only when the points a and b are equally distant from 
the surface ; or, in other words, there ivill be no rest till all 
points in the surface are on the same level. 

This fact is commonly expressed thus : ' ' Water always seeks its 
lowest level." In accordance with this principle, water flows down 
an inclined plane, and will not remain heaped up. An illustration 
of the application of this principle, on a large scale, is found in the 
method of supplying cities with water. Fig. 116 represents a modern 
aqueduct, through which water is conveyed from an elevated pond 
or river a, beneath a river 6, over a hill c, through a valley cZ, to a 
reservoir e, from which water is distributed by service-pipes to the 
dwellings in a city. The pipe is tapped at different points, and 
fountains would rise to the level of the water in the pond were it 
not for the resistance of the air and the check which the ascending 
stream receives from the falling drops. Where should the pipes be 
made stronger, on a hill or in a valley ? Where will water issue 



from faucets with greater force, in a chamber or in a basement ? 
How high may water be drawn from the pipe in the house / ? 

128. Artesian ivells, etc. — In most places, the crust of the earth 
is composed of distinct layers of earth and rock of various kinds. 

Fig. 116. 

These layers frequently assume concave shapes, so as to resemble 
cups placed one within another. Fig. 117 represents a vertical 
section exposing a few of the surface-layers of the earth's crust : 
a is a stratum of loose sand or gravel ; &, a clay-bed ; c, a stratum 

Fig. 117. 

of slate ; d, a stratum of limestone ; the whole resting on a bed of 
granite e. If you hollow out a lump of clay, and pour water into 
the cavity, you will find that the water will percolate through the 
clay very slowly. Water that falls in rain passes readily through 
the gravel a, till it reaches the clay-bed 5, where it collects. Hence 
a well sunk to the clay-bed, will fill with water as high as the water 



stands above the clay. Water also works its way from elevated 
places down between the strata of rocks. If a hole be bored 
through the slate c, water will rise above the surface of the ground 
in a fountain, seeking the level of its source on the hill ; and if bored 
still lower, through the stratum d, a still higher fountain may result. 
Such borings are called Artesian wells. Water frequently forces 
its way through fissures in the rocky strata to the surface, as at i, 
and gives rise to springs. 


I. The areas of the bottoms of vessels A, B, and C (Fig. 118) are 
equal. The vessels have the same depth, and are filled with water. 
Which vessel contains the most water ? On the bottom of which vessel 
is the pressure equal to the weight of the water which it contains ? How 
does the pressure upon 
the bottom of vessel 
B compare with the 
weight of the water 
m it? 

2. A cubic foot of 
water weighs about 
62.3 pounds or 1000 ounces. Suppose that the area of the bottom of 
each vessel is 100 square inches and the depth is 10 inches ; what is the 
pressure on the bottom of each ? 

3. Vessel A is a cubical vessel ; what is the total pressure against one 
of its vertical sides ? 

4. Suppose that vessel A were tightly covered, and that a tube 10 feet 
long were passed through a perforation in the cover so that the end 
should just touch the upper surface of the water in the vessel ; then sup- 
pose the tube to be filled with water. What additional pressure would 
each wall of the cube sustain ? 

5. Suppose that the area of the end of the large piston of a hydraulic 
press is 100 square inches ; what should be the area of the end of the 
small piston that a force of 100 pounds applied to it may produce a pres- 
sure of 2 tons ? 

[Exercises 6 to 10 mclusive should, if practicable, be actually per- 
formed in the manner directed, either by the students individually, or by 
one or two members of the class m the presence of the rest, while all 
work out the results from the data thus obtained.] 




Fig. 119. 

6. Take a glass U-tube (Fig. 119) about 40 in. high, having a stout 
rubber tube a attached, and containing mercury witli tlie surfaces at the 
same level in both arms. a. Blow into the tube ; the 
surfaces of mercury v^ill at once assume different 
levels. How will you determine the pressure which 
you exert through the air in the tube upon the mer- 
cury (the specific density of mercury being 13.59) ? 

G7. a. Suck air from a ; what happens to the mer- 
cury ? b. How will you determine the diminution of 
pressure which you produce by suction ? 

8. Take a similar tube containing water instead of 
mercury, connect it with a gas jet, and turn on the 
gas ; how would you determine how much greater (or less) its pressure 
is than that of the atmosphere ? 

9. a. Having ascertained as in Exercise 6 the greatest pressure you 
can exert by blowmg, how would you proceed to determine the greatest 
weight, placed on the piston of the seven-in-one apparatus (Fig. 120), that 
you could support, provided there were no friction and the apparatus 
were perfectly air-tight ? b. How would you estimate 
the loss in force in consequence of friction, etc. ? 

10. If the apparatus be inverted and a weight be 
hung from the piston, as in Fig. 120, how would you 
determine the greatest weight you ought to be able to 
raise by suction ? 

11. How great is the hydrostatic pressure in fresh 
water at the depth 50 feet ? 

12. Why does not a person who dives to the bottom 
of a pond feel the weight of the column of water above 
him ? 

13. Why does not the weight of the greater quan- 
tity of liquid in a coffee pot when filled cause the 
liquid to rise higher in the spout than the surface of 
the liquid in the pot ? 

14. a. A house is supplied with water by a system of pipes from a 
distant reservoir, as is customary in cities ; what data would you require 
in order to compute the pressure at any point in the pipe ? b. How 
much greater is the pressure at a point in the pipe in the cellar than at 
another point in the attic ? c. Is the pressure in the pipe the same when 
water is running from a faucet in the house as when the water is at 
rest ? Why ? 

Fig. 120. 



15. A (Fig. 121) represents a stand-pipe for furnishing the neighboring 
district with water by the action of gravity. The stand-pipe is supplied 
with water from a lake in the vicinity by means of a pumping engine. 

7n — 

Fig. 121. 

Vertical distances are represented on a scale of ^ in. =o= 50 ft. If the 
stand-pipe be filled to the level mn and the water be at rest in the 
main pipe leading from it, what pressure will the pipe sustain at points 
a, c, and d respectively ? 

Section II. 


129. Introduction. — We live at tlie bottom of an exceed- 
ingly rare and elastic aerial ocean, called the cdmosphere, 
extending to an undetermined distance into space. Every 
molecule in the gaseous ocean is drawn towards the earth's 
center by gravitation and the atmosphere is thus bound to the 
earth by this force, just as is the liquid ocean. Evidently the 
pressure in the atmosphere due to its weight increases with 
the depth ; or, since in our position we are more accustomed 
to speak of hir/ht in the atmosphere, decreases with the hight. 
The pressure does not diminish regularly with the hight as 
in an ocean of incompressible fluid. Air is very compressible 
and therefore varies in density. The lower strata of air 
sustaining the weight of air above are relatively much com- 
pressed, very dense, and elastic. The density and elasticity 
of the air diminish more rapidly than the hight above sea- 
level increases. Owing to this fact the greater part of the 



Fig. 122. 

mass of the atmosphere is within three and a half miles of 
the sea-level (see Fig. 132). Above this hight the air is 
much rarefied and vanishes, as it were, very gradually into 
empty space. 

Experiment 1. — Fill two glass jars (Fig. 122) with water, A having a 
glass bottom, B a bottom provided by tying 
a piece of sheet-rubber tightly' over the rim. 
Invert both in a larger vessel of water, C. 
The water in A does \iot feel the downward 
pressure of the air directly above it, the 
pressure being sustained by the rigid glass 
bottom. But it indirectly feels the pressure 
of the air on the surface of the water in the 
open vessel, and it is this pressure that sustains 
the water in the jar. But the rubber bottom 
of the jar B yields somewhat to the down- 
ward pressure of the air, and is forced inward. 
Experiment 2. — Fill a glass tube, D, with water, keeping the lower 
end in a vessel of water, and the upper end 
tightly closed with a finger. Why does not 
the water in the tube fall ? Remove your 
finger from the closed end. Why does the 
water fall ? 

Experiment 3. — Fill (or partly fill) a 
tumbler with water, cover the top closely 
with a card or writing-paper, hold the 
paper in place with the palm of the hand, 
and quickly invert the tumbler (Fig. 123). 
Why does not the water fall out ? 

Experiment 4. — Force the piston A (Fig. 
124) of the seven-in-one apparatus quite to 
the closed end of the hollow cylinder, and 
close the stop-cock B. Try to pull the piston out again. Why do you 

not succeed? Hold the apparatus in 
various positions, so that the atmosphere 
may press down, laterally, and up, 
against the piston. Do you discover 
any difference in the pressure which it 
receives from different directions ? 

Fig. 123. 


130. How atmospheriG pressure is measured. 

Experiment 5 (preliminary), —Take a U-sliaped glass tube (Fig. 125), 
half fill it with water, close one 
end with a thumb, and tilt the 
tube so that the water will run 
into the closed arm and fill it; 
then restore it to its original ver- 
tical position. Why does not the 
water settle to the same level in 
both arms ? 


Fig. 125. 


126 represents a U-sliaped glass tube closed at one 
end, 34 inches in higlit, and with, a bore of 
1 square inch section. The closed arm hav- 
ing been filled with mercury, the tube is 
placed with its open end upward, as in the 
cut. The mercury in the closed arm sinks 
about 2 inches to A, and rises 2 inches in 
the open arm to C ; but the surface A is 30 
inches higher than the surface C. This can 
be accounted for only by the atmospheric 
pressure. The column of mercury B A, con- 
taining 30 cubic inches, is an exact coun- 
terpoise for a column of air of the same 
diameter extending from C to the upper 
limit of the atmospheric ocean, — an un- 
known hight. 

The weight of the 30 cubic inches of mer- 
cury in the column B A is about 14.7 pounds. 
Hence the weight of a column of air of 1 square-inch section, 
extending from the surface of the sea to the upper limit of 
the atmosphere, is about 14.7 pounds. But in fluids gravity 
causes equal pressure in all directions. Hence, at the level 
of the sea, all bodies are pressed upon in all direetions by the 
atmosphere, ivith a force of about 14,7 pounds per square inch, 
or about one ton per square foot. 

Fig. 126. 



A pressure of 15 pounds per square inch is quite generally 
adopted by engineers as a unit of gaseous pressure, and is 
called an atmosphere. Physicists, however, generally measure 
pressure in terms of cm or mm of mercury at 0° C. ; that is, 
the hight in myii of mercury that the pressure of the atmos- 
phere sustains in the tube. 

Fig. 127. 

131. The barometer. — The hight of the col- 
umn of mercury supported by atmospheric pres- 
sure is quite independent of the area of the 
surface of the mercury pressed upon ; hence the 
apparatus is more conveniently constructed in 
the form represented in Pig. 127. 

A straight tube about 34 inches long is closed 
at one end and filled with mercury. The tube is 
inverted, with its open end tightly covered with 
a finger, and this end is inserted into a vessel of 
mercury. When the finger is withdrawn the 

Fig. 128. 



mercury sinks until there is equilibrium between the down- 
ward pressure of the mercurial column AB and the pressure 
of the atmosphere. The empty space at the top of the tube 
is called a Torricellian ^ vacuum. An apparatus designed to 
measure atmospheric pressure is called a barometer (pressure- 
measurer). A common cheaj^ form of barometer is repre- 
sented in Pig. 128. Beside the tube and near its top is a 
scale graduated in inches or centimeters, indicating the hight 
of the mercurial column. For ordinary purposes this scale 
needs to have a range of only three or four inches, so as to 
include the maximum fluctuations of the column.^ 

132, The Fortin harometer. — We will suppose the scale of a 
barometer to be fixed so as to indicate correctly the hight of the 
surface of mercury in the tube above that in the 
cistern at a time, for instance, when this distance 
is 30 inches. A point on the surface of the mercury 
in the cistern in this case is called technically the 
zero point. Now should the mercury fall in the 
tube to 29 and the mercury in the cistern remain 
at zero, then the scale reading would indicate 
correctly the barometric hight ; but the mercury 
does not remain at zero, but rises a little (less as 
the diameter of the cistern is greater) , consequently 
the scale-reading is too great. When the mercury 
is higher in the tube than 30 all the readings will 
be too small. Evidently, then, the mercury in the 
cistern must be brought to zero at every obser- 
vation in order to eliminate this error. This is 
easily accomplished with the Fortin barometer. 
The bottom of the cistern of this barometer is 
pliable leather resting on a thumb-screw A (Fig. 
129). Projecting from the tube inside of the cistern 
is a little pointer B of colored glass. The lower end 
of this pointer, called the fiducial point, corresponds 
to the zero point. The level of the mercury in the 
cistern must be set to this point by raising or fig. 129. 

1 The first barometer was constructed by Torricelli, a Florentine, in 1643. 

2 At the Central Station in Boston, Feb. 8, 1895, the mercury fell to 28.61 in., the 
lowest on record at this station. The highest point attained at this station Avas 30.97 
in, on Dec. 1, 1887, and Dec. 31, 1889. 




lowering the cistern base by the adjusting screw, before taking a 
reading. A sliding piece C, Fig. 130, furnished with a vernier i 
can be slid along the tube so as to enable one to read with great 

In refined scientific researches it is necessary to make suitable 
allowances for expansion and contraction of the mercury attending 
changes in temperature, hence a very sensitive ther- 
mometer is attached to the barometer. Also allowances 
for capillary depression must be made. 

133. The aneroid barometer. — The aneroid (without 
moisture) barometer employs no liquid. It contains a 
cylindrical box, D (Fig. 131), having a very flexible top. 
The air is partially exhausted from within the box. The 
varying atmospheric pressure causes this top to rise and 
sink much like the chest of a man in breathing. Slight 
movements of this kind are communicated by means of 
multiplying-apparatus (apparatus by means of which a 
small movement of one part causes a large movement of 
another part) to the index needle A. The dial is gradu- 
ated to correspond with a mercurial barometer. The 

observer turns the button C and brings the brass needle B over the 

black needle A, and at his next 

observation any departure of the 

latter from the former will show pre- 
cisely the change which has occurred 

between the observations. 

The aneroid can be made more 

sensitive (i.e. so as to show smaller 

changes of atmospheric pressure) 

than the mercurial barometer. 

Owing to this as well as to its con- 
venient size and portability, the 

aneroid has become quite popular. 

Unfortunately, however, it does not 

preserve its accuracy for a great 

length of time, hence it must be adjusted from time to time to a 

standard mercurial barometer. 

Fig. 130. 

Fig. 131. 

1 For construction and methods of using verniers the student is referred to 
Pickering's " Manual of Physical Manipulations," or Stewart and Gee's " Elementary 
Practical Physics." 


134. Standard ])vessiire. — Many physical operations require 
a standard pressure for reference. The standard generally 
adopted is the pressure exerted by a column of pure mercury 
at 0° C. and 76 cm. (29.922 inches) high, which is about the 
average hight of the barometric column at sea-level in latitude 
45"". The pressure corresponding to this hight is 1033.3 
grams per square centimeter or 14.69 pounds per square 

135. Barometric measurement ofhights. — Since atmospheric 
pressure varies with the hight above sea-level, it is evident 
that changes in elevation may be determined from changes 
of pressure as indicated by the barometer. In other words, 
the hight of a mountain may be ascertained from barometric 
readings made on the summit and at sea-level. Such deter- 
minations are more reliable for moderate elevations, as there 
are elements of greater or less uncertainty in measuring great 
hights. For moderate hights the barometric column falls at 
a very nearly uniform rate of one inch for every 900 feet of 

If a mercurial barometer stand at 760 mm. on the floor, the 
same barometer on the top of a table 1 m. high should stand 
at a hight of 759.91 mm., a change scarcely perceptible. The 
aneroid is, however, sometimes made so sensitive that the 
change of pressure experienced in this short distance is ren- 
dered quite perceptible. 

The shading in Fig. 132 is intended to indicate roughly the 
variation in the density of the air at different elevations above 
sea-level. The figures in the left margin show the hight in 
miles ; those in the first column on the right, the corresponding 
average hight of the mercurial column in inches ; and those 
in the extreme right, the density of the air compared with its 
density at sea-level. 

It is calculated that if an opening could be made in the 
earth 35 miles in depth below the sea-level, the density of the 



air at the bottom would be 1,000 times that at sea-level, so 
that water would float in it. 

00 30000 



Fig. 132. 

If the aerial ocean were of uniform density, and of the same 
density that it is at sea-level, its depth would be a little less 
than five miles. Only a few peaks of the Himalayas would 
rise above it. 

136. The barometer in meteorology.'^ — The barometer is some- 
times called a "weather-glass," chiefly because its scale frequently 
bears the words fair., rainy, storm, etc. These words are very 
objectionable, since they are totally wrong from a meteorological 
point of view. To form a forecast of the weather of much value, a 

1 The following works will he found useful to students of meteorology: "Ele- 
mentary Meteorology," by W. M. Davis ; " Instructions in tlie Use of Meteorological 
Instruments," " Elementary Meteorology," " Weather Charts and Storm Warnings," 
by H. H. Scott; "Weather Casts and Storm Prevision," by K Strachan; and "A 
Treatise on Meteorological Instruments," by Negretti and Zambra. 


barometer, a thermometer, and a hygrometer must be consulted, 
and one must be familiar with the laws which govern the relations 
between atmospheric pressure, temperature, moisture, etc. In 
forming a judgment of forthcoming weather, the point at which the 
mercury stands should not be so much regarded as whether it is 
rising or falling. The following general rules may not be amiss, 
though even these have many exceptions : — 

1. A steady barometer at about its mean hight, with a seasonable 
temperature, and dry air, indicates a continuation of fine weather. 

2. A rise from this point indicates decidedly fine and dry ; a fall 
indicates rain or higher wind. 

3. A gradual rise or fall indicates a less immediate change than 
a more rapid motion of the mercury. 

Eluctuations in barometric pressure are of hourly occurrence. 
Some of the many conditions which influence the atmospheric pres- 
sure are the following -. (1) Temperature. A rise of temperature 
tends to diminish the air-pressure. In general the barometer falls 
as the thermometer rises. Heat expands the air, causing a lateral 
flow away from the heated areas. (2) Humidity. Moist air is 
lighter than dry air having an equal pressure. (3) Currents in the 
atmospheric ocean. On weather charts, lines called isobars are drawn 
through places having the same pressure. Frequently the isobar 
indicating the lowest pressure encloses an area more or less circular. 
This is called a "center of depression." The surrounding air tends 
to flow into it from all sides. The greater the difference of pressure 
between the two places, i.e. the steeper the barometric gradient, 
the more rapid the flow. The direction of the wind, however, is so 
modified by the rotation of the earth that the flow is not directly 
toward the center, but spirally toward it, the motion in the northern 
hemisphere being opposite to the direction in which the hands of a 
watch move. 



Section III. 


137. Boyle's (or Mariotte^s) Law. 

Experiment 1. — Take a bent glass tube (Fig. 133), the short arm being 
closed, and the long arm, which should be at least 34 
inches (85 cm.) long, being open at the top. Pour 
mercury into the tube till the surfaces in the two 
arms stand at zero. Now the surface in the long 
arm supports the pressure of an atmosphere. There- 
fore the pressure of the air enclosed in the short arm, 
which exactly balances it, must be about 15 pounds 
to the square inch. Next pour mercury into the long 
arm till the surface in the short arm reaches 5, or till 
the volume of air enclosed is reduced one half, when 
it will be found that the hight of the column A C is 
just equal to the hight of the barometric column at 
the time the experiment is performed. It now ap- 
pears that the pressure of the air in A B balances the 
atmospheric pressure, plus a column of mercury A C 
which is equal to another atmosphere ; .-. the pres- 
sure of the air in A B = two atmospheres. But the 
air has been compressed into half the space it for- 
merly occupied, and is, consequently, twice as dense. 
If the length and strength of the tube would admit of 
a column of mercury above the surface in the short 
arm equal to twice A C, the air would be compressed 
into one third its original bulk ; and, inasmuch as it 
would balance a pressure of three atmospheres, its 
pressure would be increased threefold. 

This experiment may be conducted in a more sci- 
entific manner, as follows : — 

Experiment 2, — Let the mercury be at the same 

level, AB (Fig. 134), in both arms of the tube. The 

body of air to be experimented with is in the short arm between A and 

C. The dimensions of this body can vary only in hight ; hence its 

hight, H, may represent its volume. Measure H, i.e. the distance 

Fig. 133. 



between A and C, and regard the number of inches (or centimeters) 
as representing the volume, V. Its pressure, P, evidently is the same as 
that of the atmosphere at the time of experimenting. 
Consult a barometer, and ascertain the hight of the baro- 
metric column ; let this hight represent P. Pour a little 
mercury into the tube ; the mercury rises to Ai and Bi. 
Measure from Ai to C ; this number represents the vol- 
ume, Fi, of the body of gas now. Measure the vertical 
distance between Ai and Bi ; this number represents the 
increase in pressure, which, added to P, will give its 
present pressure, Pi. [— Bs 

Now pour more mercury into the long arm, so that it 
win rise to some such points as A2 and Bg. Determine 
as before the new volume, Fo, and new pressure, P2. 
So continue to add mercury a third, and a fourth time, 
and get new values for the volume, F3, and F4, and 
for the pressure, P3, and P4. Arrange the results as 
follows : — 

F = 


P = 

Pi = 

P2 = 


V X P = 
Vi xp,= 






Fig. 134. 

It will be found that the series of products in the last 
column are approximately equal (due allowance being made 
for errors in measurement, etc.) ; consequently F varies 
inversely as V. Hence the law : — 

The volume of a body of gas at a constant temperature varies 
inversely as its pressure, density, and elasticity. 

For many years after the announcement of this law, first 
by Boyle and a little later by Mariotte, it was believed to 
be rigorously correct for all gases ; but more recently, more 
precise experiments have shown that it is approximately but 
not rigidly true for any gas, that the departure from the law 
differs with different gases, and that each gas possesses a 
special law of compjressibility. There is a limit beyond which 
this law does not hold. This limit is soonest reached with 
those gases, like carbon-dioxide, chlorine, etc., that are most 




readily liquefied by pressure. A gas is nearer perfect, or 
conforms more nearly to Boyle's law, in proportion as it is at 
a greater distance, as regards both temperature and pressure, 
from its liquefying point. When a gas is near the critical 
state its density increases more rapidly than its elasticity.^ 

138. Manometer, or pressure gauge. — The manometer, an instru- 
ment for measuring the pressure of a gas or vapor contained in a 
closed vessel, under considerable pressure, illustrates the application 
of Boyle's law to a practical purpose. 

This instrument consists, as in Fig. 135, of a bent tube A B closed 
at one end a, and containing within the space 
Aa a quantity of air, which is cut off from 
external communication by a column of mer- 
cury. The apparatus is so constructed that 
when the pressure on B is equal to that of an 
atmosphere, the mercury stands at the same 
hight in both branches. But if the pressure 
increase, the mercury is forced into the left 
branch, so that the air in that branch is com- 
pressed, and its elasticity proportionately in- 
creased. The pressure of the gas exerted at B 
is then equal to the pressure of the compressed 
air, together with that of a column of mercury 
m n equal to the difference of level of the liquid in the two branches. 
This pressure is expressed in atmospheres on the scale ab. 

139. Elasticity of gases. — The elasticity of all fluids is 
perfect. By this is meant, that the force exerted 
in expansion is, except within certain limits 
rarely reached, equal to the force used in com- 
pression; and that, however much a fluid is 
compressed, it will always completely regain 
its former bulk when the pressure is removed. 

Hence the barometer, which measures the com- ^L -t^ 

pressing force of the atmosphere, also measures ^^^^"^^T^^"^ 

at the same time the elasticity of the air. 

A so-called vacuum gauge (Fig. 136) is simply a short mer- 

1 The student will find this subject admirably treated in Daniell, pp. 204, 222, etc. 

Fig. 135. 



cury barometer, — short because it is seldom required to 
make measurements except in tolerably high vacua, where 
the mercurial column is correspondingly low. For instance, 
this apparatus, placed under the receiver of an air-pump 
from which air is exhausted, will measure the elasticity of 
the air in the receiver. This known, the degree of exhaustion 
is readily determined. 

Experiment 3. — Force the piston of the seven-in-one apparatus two 
thirds of the way into the cylinder, and close the aper- 
ture. Support the apparatus on blocks, with the piston 
upwards, and place the whole under the receiver of an 
air-pump. Exhaust the air from the receiver ; the outside 
pressure of the air being partially removed, the unbalanced 
pressure of the air enclosed within the cylinder will cause 
the piston to rise. 

Experiment 4. — Take a glass tube (Fig. 137) having a 
bulb blown at one end. Nearly fill it with water, so 
that when it is inverted there will be only a bubble of air 
in the bulb. Insert the open end in a glass of water, place under a 
receiver, and exhaust. Nearly all the water will leave the bulb and tube. 
Why ? What will happen when air is admitted to the 
receiver ? 

140. Suction. — Liquids are said to be raised 
by the "force of suction," which seems to imply 
that a lifting force acting from above ^:>?6Z^s- the 
liquid up. Does this explain suction ? 

Fig. 137, 

Experiment 5. — Fill a glass U-tube having unequal 
arms (Fig. 138) with water to the level c h. Close the end 
6 with a finger, and try to suck the liquid out of the tube. You find it 
impossible. Eemove the finger from 6, and you can suck the liquid out 
with ease. Why ? 



Sectiox ly. 


141. The aiT-puin2J. — The air-pump 


FlCx. 139. 

is used to rarefy air 
in a closed vessel. 
Fig. 139 will serve 
to illustrate its 
operation. E is a 
glass receiver within 
which the air is to 
be rarefied; B is a 
hollow cylinder of 
brass, called the 
picmp -barrel ; the 
plug P, called a ins- 
ton, is fitted to the 
interior of the bar- 
rel, and can be moved up and down by the handle H ; *' and t 
are valves. A valve acts on the principle of a door intended 
to open or close a passage. If you walk against a door on 
one side, it opens and allows you to pass ; but if you walk 
against it on the other side, it closes the passage, and stops 
your progress. Suppose the piston to be in the act of 
descending ; the compression of the air in B closes the valve 
t, and opens the valve s, and the enclosed air escapes. After 
the piston reaches the bottom of the barrel, it begins its 
ascent. This would cause a vacuum between the bottom of 
the barrel and the ascending piston (since the unbalanced 
pressure of the outside air immediately closes the valve s), 
but the pressure of the air in the receiver E, opens the valve t 
and fills this space. As the air in E expands, it becomes 
rarefied and exerts less pressure. The external pressure of 
the air on E, being no longer balanced by the pressure of 



the air within, presses the receiver firmly upon the plate L. 
Each repetition of a double stroke of the jjiston removes a 
portion of the air remaining in R. The air is removed from 
E by its own expansion. However far the process of exhaus- 
tion may be carried, the receiver will always be filled with 
air, although it may be exceedingly rarefied. The operation 
of exhaustion is practically ended when the pressure of the 
air in R becomes too feeble to lift the valve t, unless the 
apparatus be so constructed that the valves are opened and 
closed by mechanical action. It is obvious that if s and t 
opened downward instead of upward, then as the piston is 
raised and depressed, air would be compressed in R. A con- 
denser is merely a pump with its valves reversed, and is used 
to condense air. 

142. The mercury air-pump. — In recent years the so-called 
mercury air-pump has largely displaced the pump described 
above, since it is capable of producing a much greater rare- 
faction. In brief, it makes use of the Torricellian vacuum, 
such as is formed in the top of a barometer tube. On account 
of its simplicity, the Geissler pump, the first of the kind 
invented, is chosen for illustration. A (Fig. 140) is a glass 
tube more than thirty- e 

four inches long, having ^ ^M^^ 

a globe-like enlargement 
B of about a liter capac- 
ity. Above this globe 
leads a tube containing a 
three-way stop-cock and 
a branch tube D. By 
means of this stop-cock 
B may be placed in com- 
munication either through 
D with the atmosphere, 
as shown in M (Fig. 141), or through E with the receiver to 

Fig. 140. 

Fig. 141. 


be exhausted, as shown in N. Connected with, the lower end 
of A by means of a thick rubber tube is a vessel G containing 
mercury. The pump is operated as follows : C is turned as 
in M, G- is raised so that mercury will flow from it into B 
and fill it, the air escaping through D. Then the stop-cock is 
placed as in 'N, and G- is lowered so as to allow the mercury 
to flow back into it. A Torricellian vacuum would be formed 
in B were it not in communication through E with the 
receiver. As it is, the air in this space expands and fills B, 
and is thus to this extent rarefied. By a sufficient number 
of repetitions of this process, a very high vacuum is obtaina- 
ble. There are many modifications of this pump, in some of 
which the stop-cock is dispensed with, and consequently the 
trouble of operating it is avoided. 

With the common pump a vacuum of a millimeter of mer- 
cury is considered exceedingly good ; but with a mercury pump 
it is easy to obtain a vacuum of .00076 of a millimeter, which 
represents about one millionth the normal pressure of the 
atmosphere. "Eood in 1881 succeeded in obtaining vacua 
as high as a three hundred-millionth of an atmosphere." — 

Section V. 


143. Construction and operation of the siphon. — A siphon 
is an instrument used for transferring a liquid from one vessel 
to another over an elevation through the agency of atmo- 
spheric pressure. It consists of a tube of any material (rub- 
ber is often most convenient) bent into a shape somewhat 
like the letter U. To set it in operation, fill the tube with 
a liquid, stop each end with a finger or cork, place it in the 
position represented in Fig. 142, remove the stoppers and 
the liquid will flow out at the orifice o. Why? The up- 


ward pressure of the atmosphere against the liquid in the 
tube is the same at both ends ; hence these two forces are 
in equilibrium. But the downward 

pressure of the column of liquid ah T ~Jr\ "i" 

is greater than the downward pres- ^' ^^P \\~ ^^ 

sure of the column d c ; hence equi- ^^S| \\ j 

librium is destroyed and the move- \\ i 

\\ j 

ment is in the direction of the &\—ih 

greater (i.e. the unbalanced) force. 

The unbalanced force which causes the flow is equal to the 
downward pressure of the column e h. 

If one end of the tube filled with liquid be immersed in a 
liquid in some vessel, as in A (Fig. 143), and the other end 
be brought below the surface of the liquid in the vessel and 
the stoppers be removed, the liquid in the vessel will flow out 
through the tube as long as the distance e h remains greater 
than zero. 

If one of the vessels be raised a little, as in C, the liquid will flow 
from the raised vessel, till the surfaces in the two vessels are on the 
same level. The remaining diagrams in this cut represent some 
of the great variety of uses to which the siphon may be put. D, E, 
and F are different forms of siphon fountains. In D, the siphon 
tube is filled by blowing in the tube /. Explain the remainder of 
the operation. A siphon of the form G is always ready for use. It 
is only necessary to dip one end into the liquid to be transferred. 
Why does the liquid not flow out of this tube in its present con- 
dition ? H illustrates the method by which a heavy liquid may be 
removed from beneath a lighter liquid. By means of a siphon a 
liquid may be removed from a vessel in a clear state, without 
disturbing sediment at the bottom. I is a Tantalus cup. A liquid 
will not flow from this cup till the top of the bend of the tube is 
covered. It will then continue to flow as long as the end of the 
tube is in the liquid. The siphon J may be filled with a liquid that 
is not safe or pleasant to handle, by placing the end j in the liquid, 
stopping the end jt, and sucking the air out at the end I till the 
lower end is filled with the liquid. 



Gases heavier than air may be siphoned like liquids. Vessel o 
contains carbonic acid gas. As the gas is siphoned into the vessel p, 
it extinguishes a candle-flame. Gases lighter than air are siphoned 
by inverting both the vessels and the siphon. 

Fig. 143. 

The siphon cannot elevate liquids, it can merely transfer 
liquids to places of lower level. It is apparent that the 
pressure of the air drives the liquid through the siphon, and 
that a siphon would be inoperative in a vacuum. Obviously 
the hight of the bend to which different liquids can be raised 
varies with their respective densities. Thus the hight to 
which mercury can be raised is the barometric hight, while 
water may be raised 13.6 times as high. 



144. Lifting or suction puinp. — The common lifti7ig-pump 
is constructed like the barrel of an air-pump. Fig. 144 rep- 
resents the piston B in the act of rising. As the air is rarefied 
below it, water rises in consequence of atmospheric pressure 
on the water in the well, and opens the lower valve D. 

Fig. 145. 

Fig. 146. 

Fig. 144. 

Atmospheric pressure closes the upper 
valve C in the piston. When the piston is 
pressed down (Fig. 145), the lower valve 
closes, the upper valve opens, and the 
water between the bottom of the barrel 
and the piston passes through the upper 
valve above the piston. When the piston is raised again 
(Fig. 146), the water above the piston is raised and discharged 
from the spout. 

The liquid is sometimes said to be raised in a lifting-pump 
by the "force of suction." Is there such deforce? 

Calling the specific density of mercury 13.6 and disregarding 
the vapor pressure of mercury (.02 mm at 20° C), a water 
barometer would be 76 X 13.6 = 1033.6 cm high when the 
mercury barometer stands at 76 cm, provided there were no 
pressure exerted by the vapor of water. But at 20° C. the 



pressure of water vapor is 1.74 cm of mercury = 1.74 X 13.6 
(=^ 23.7 cm) centimeters of water. Therefore the hight of the 
water barometer would be 1033.6 — 23.7 = 1009.9 cm, and this 
is the limit of the hight to which water can be raised by the 
pressure of air in a suction pump under these conditions.^ 

145. Suction and foixe-pumjj comhined. — In this pump the 
ordinary piston with valve and leather washers is replaced 
by a solid cylinder of metal, B (Fig. 147), called the plunger. 

This passes through a stuffing box 
D, in which it fits air-tight. Valves 
opening upward and outward are 
placed at A and C respectively. 
When the plunger is raised, A 
opens and C closes, and water is 
raised into the barrel by atmos- 
pheric pressure. When the plunger 
descends, A closes and C opens, and 
the water is forced up through the 
pipe E to a hight dependent on the 
pressure brought to bear upon it 
through the plunger. An air-dome 
F is usually connected with these 
pumps to regulate the pressure so 
as to give through the delivery pipe a very steady stream. 
This dome contains air. When the plunger descends it forces 
water violently into E and thus tends to produce a severe 
strain in the pipe. But the water enters the dome, compresses 
the elastic air within, and thus the shock is largely reduced. 
As soon as the down stroke of the piston ceases, the valve C 
closes, and the compressed air in the dome forces the water 
out through E in a continuous stream. 

Fig. 147. 

1 At 50° the limit would be 1033.6 - (9.2 x 13.6) = 908.4 cm. At 100° the hight would 
be 1033.6 — (76 x 13.6) = cm, the pressure of water vapor being equal to the pressure 
of the atmosphere. 


Section VI. 


146. Whi/ a solid is huoijecl up by a fluid, and ivith hoiv 
great a force it is buoyed up. — Suppose dob a (Fig. 148) to 
be a cubical block of marble immersed in a liquid. It is 
obvious tliat the downward pressure upon the surface da is 
equal to the weight of the column of liquid 
edao. The upward j)i'essure on the surface '^ ^ ^ 

c^ is equal to the weight of a column of 
liquid ecbo. The difference between the 
upward pressure against cb and the down- 
ward pressure on da, is the weight of a 
column of liquid ecbo less the weight of a 
column of liquid edao, which is a column of 
liquid deb a (ecbo — eda.o^= deb a). But -p^^ -^^g 

a column of liquid deb a has precisely the 
volume of the solid submerged. Therefore, a solid is buoyed 
up) by a fluid in consequence of the unequal pressures upon its 
top and bottom at their different depths, and tlie amount of the 
buoyancy is the iveight of a volume of that fluid equal to the 
volume of the immersed solid. 

This principle ^ may be thus stated : a solid immersed in a- 
fluid is buoyed upj by a force equal to the iveight of the fluid 
displaced. The difference between the weight of a body and 
the buoyant force of a fluid in which it is submerged may be 
called the effective iveight of the body in that fluid. 

Experiment 1. — Suspend from one arm of a balance beam a cyhn- 
drical bucket A (Fig. 149), and from the bucket a sohd cyhnder whose 
volume is exactly equal to the capacity of the bucket ; in other words, the 
latter would just fill the former. Counterpoise the bucket and cylinder 
with weights. 

1 This principle is commonly called the Archimedes principle from the name of 
the discoverer (287 to 212 B.C.). 



until the 

beneath the cylinder a tumbler of water, and raise the tumbler 

cylinder is completely submerged. The buoyant force of the 

water destroys the equilibrium. Pour water 

I ' into the bucket ; when it becomes just even 

I full, the equilibrium is restored. 

Now it is evident that the cylinder 
immersed in the water displaces its own 
volume of water, or just as much water 
as fills the bucket. But the bucket full of 
water is just sufficient to restore the weight 
lost by the submersion of the cylinder. 
What principle does this experiment illus- 
trate ? 

A floating body, as a cork on water, 
has no effective weight. It sinks 
until it disjjlaces a weight of the fluid 
equal to its own weight, or until it 

Fig. 149. 

reaches a depth where the upivard i^ressure of the fluid is equal 
to its own weight. 

Experiment 2. — Place a baroscope (Fig. 150), consisting of a scale- 
beam, a small weight, and a hollow brass sphere, under the receiver of an 
air-pump, and exhaust the air. In the air the weight and sphere balance 
each other; but when the air is removed, the 
sphere sinks, showing that in reality it is heavier 
than the weight. In the air each is buoyed up by 
the weight of the air it displaces; but as the 
sphere displaces more air, it is buoyed up more. 
Consequently, when the buoyant force is with- 
drawn from both, their equilibrium is destroyed. 

Ordinary weighing conducted in the air 
consists, therefore, in a comparison of 
effective weights in that fluid. The ahso- 
lute iveight is, evidently, the effective 
weight plus the weight of the excess of air 

Fig. 150. 

displaced by the body over that displaced by the weights, or 
it is the weight of the body in a vacuum. 


The density of the atmosphere is greatest at the surface of 
the earth. A body free to move cannot displace more than 
its own weight of a fluid ; therefore a balloon, which is a 
large bag filled with a gas about fourteen times lighter than 
air at the sea-level, will rise till the weight of the balloon, 
together with its car and cargo, equals the weight of the air 

Section YII. 

density and specific density. 

147. Meaning of the terms and their relation to each other. 

— The density of a substance at any temperature is the mass of 

a unit volume of the substance at that temperature. Thus, the 

density of water at 4° C. is one gram per cubic centimeter, and 

the density of cast iron at the same temperature is about 7.12 

grams per cubic centimeter. The mean density of a body is 

found by dividing its mass by its volume. Thus if the mass 

of a body be 30 grams and its volume be 6 cubic centimeters, 

its mean density is (30 -f- 6 =) 5 grams per cubic centimeter. 

The dimensional equation for density (D) is [D] = ^=— r^ = [M] [L] — ^. 

As every substance has a special density of its own, the 
special (specific) density of any substance is most conveniently 
measured by comparison with the density of some substance 
chosen as a standard. 

The specific density of a substance at any temperature is the 
ratio of its density at that temperature to the density of some 
standard., i.e. it is the ratio between the masses of equal volumes 
of a given sidystance and of some standard substance ; or, since 
weight at the same place is proportional to mass, it is the 
ratio betiveen the weights of equal volumes of the substance and 
of the standard. The latter ratio is commonly known as the 
specific gravity of the substance. 


It must be carefully borne in mind that density represents 
a definite number of units of mass (i.e. the number contained 
in some unit of volume) and is therefore a concrete number ; 
while specific density is simply a ratio, and hence an abstract 

The standard adopted in scientific work for solids and 
liquids is the density of distilled water at 4° C. In the metric 
system the number which expresses the numerical value of 
the density of a substance and the number which expresses 
the specific density of the substance are identical ; the latter, 
being a ratio, is, of course, an abstract number. Thus the 
density of water being one gram per cubic centimeter at 4° C, 
and the density of cast iron at the same temperature being 
7.12 grams per cubic centimeter, the ratio of the latter to 
the density of the former (i.e. the specific density of iron) 
is 7.12, which is the same as the numeric of the density of 

148. Foi'mulas for specific density. — Let D represent the 
density of any given substance, and D' the density of water, 
and let W and W represent respectively the weights of equal 
volumes of the same substances ; then, by definition, 

Density of given substance D 

' Density of water D' *' 

/ON Weight of a given volume of the substance W 

Weight of equal volume of water W 

149. Ex2Jerimental methods of finding the specific densities 
of substances. — (1) Solids. — The Principle of Archimedes is 
commonly applied to determine the specific densities of solids. 

Experiment 1. — From a hook beneath a scale-pan (Fig. 151) suspend 
by a fine thread a small portion of the solid substance whose specific 
density is to be found, and weigh it, while dry, in the air. Then immerse 
the body in a tumbler of water (see that it is completely submerged), and 
weigh it in water. The loss of weight in water is evidently W, i.e. the 



weight of the water displaced by the body ; or, in other words, the weight 
of a body of water having the same volume as that of the specimen. 
Apply the formula (2) for finding the specific 

Experiment 2. — Take a piece of sheet lead 
one inch long and one-half inch wide, weigh 
it in air and then in water, and find its loss 
of weight in water. Weigh in air a piece of 
cork or other substance that floats in water ; 
then fold the lead-sinker, place it astride the 
string just above the specimen, completely 
immerse both, and find their combined weight 
in water. Subtract their combined weight in 
water from the sum of their weights in air ; 
this gives the weight of water displaced by both. Subtract from this the 
weight lost by the lead alone, and the remainder is W^, i.e. the weight 
of water displaced by the cork. Apply formula (2), as before. 

Prepare blanks, and tabulate the results of the experiments as follows : 

Fig. 151. 

Name of Substance. 

W in 



Sp. D. 




■ 0.66 



When the result obtained differs from that given in the table of 
specific densities (Appendix), the differ- 
ence is recorded in the column of errors 
(E). The results recorded in the column 
of errors are not necessarily real errors ; 
they may indicate the degree of impurity, 
or some peculiar physical condition of the 
specimen tested. 

(2) Liquids. 

Experiment 3. — Take a bottle that 

holds when filled a certain (whole) num- 

FiG. 152. ber of grams of water, e.g. 100 g, 200 g. 



etc. Fill the bottle with the liquid whose specific density is sought. 
Place it on a scale-pan (Fig. 152), and on the other scale-pan place a piece 
of metal a which is an exact counterpoise for the bottle when empty. 
On the same pan place weights 6, until there is an equilibrium. The 
weights placed in this pan represent the weight W of the liquid in the 
bottle. Apply formula (2). The W (i.e. the 100 g, 200 g, etc.) is usually 
etched on bottles constructed for this purpose. 

Experiment 4. — Take a pebble stone {e.g. quartz) about the size of a 
large chestnut ; find its loss of weight {i.e. W) in water; find its loss of 
weight {i.e. W) in the given liquid. Apply formula (2). 

Experiment 5. — Insert the glass tube B (Fig. 153) into a tumbler of 
the liquid whose specific density is sought, and tube C into a tumbler 
of water. Attach a stop-cock to the rubber connector D, 
raise the liquids a little way in their respec- 
tive tubes by suction, close the stop-cock, and 
measure the hights {m and n respectively) of the 
columns in B and C above the surfaces of the 




liquids in the tumblers below 


specific density of the liquid in B. 

150. The densmietev. — The principle 
of the densimeter (commonly but inappro- 
priately called the hydrometer^ is based 
upon two facts : (1) a floating solid sinks 
until it displaces its own weight of the 
liquid in which it floats ; (2) the volumes 
of tw^o liquids displaced by the same float- 
ing solid vary inversely as their densities. 

Experiment 6. — Take a prism of paraffined 
wood (Fig. 154) -J- inch square and 5 inches long, 
with a quarter-inch scale on one of its faces. It should be so loaded as 
to assume a vertical position and sink just 4 inches when placed in water. 
It displaces therefore a volume, V, (^ X ^ X 4 = 1 cu. in.) of water. 
Place it in some liquid whose specific density is sought. It displaces a 


Then — = the specific density of the given 






Fig. 153. 

Fig. 154. 

volume, Y\ of this liquid. 

liquid. This experiment illustrates the principle on which the densimeter 
is based. 



Fig. 155. 

Instead of a prism of wood, a glass tube A (Fig. 155) terminating in a 
bulb containing shot or mercury is generally used. It has a scale of 
specific densities on the stem, so that no computation is necessary. The 
experimenter merely places it in the liquid to be 
tested, and reads the specific density at that point, B, 
which is at the surface of the liquid. 

(3) Gases. 

The specific density of a gas is found by the 
application of the same principles as those em- 
ployed in determining that of a liquid, but the 
operation is attended with peculiar difficulties. 
Air is extracted from a light capacious glass or 
copper globe by means of a good air-pump, and 
the empty vessel is weighed. It is then put in 
communication, by means of a stop-cock, with 
a reservoir containing the gas to be tested, which 
must be perfectly pure and dry. Gas is allowed 
to enter the globe slowly until the pressure is 
the same as the atmospheric pressure outside, and then it is allowed 
to stand until the gas acquires the temperature of the air outside. 
The vessel thus filled with gas is again weighed ; its gain in weight is 
the weight of the gas (W) which fills the globe. The globe is again 
exhausted and filled, at the same temperature and pressure, with 
a gas which is employed as a standard. The weight of the gas now 
filling the globe is found. Then the weight of the given gas divided 
by the weight of an equal volume of the standard gas at the same 
temperature and pressure is the specific density of the given gas. 

Eor many purposes it is most convenient to employ hydrogen 
gas — the lightest gas — as a standard for gases. Then assuming 
the density of hydrogen to be 1, that of air is 14,47, oxygen 16, etc. 
A cubic centimeter of hydrogen at 0° C. and at the barometric 
pressure of 760 mm weighs at Paris 0.0000895682 g, and a cubic 
centimeter of dry pure air under the same conditions weighs 
0.0012932 g. 

151. Miscellaneous experiments. 

Experiment 7. — Find the volume of an irregular shaped body, e.g. 
a stone. Find its loss of weight in water. Remember that the loss of 
weight is precisely the weight of the water it displaces, and that the 
volume of one gram of water is one cubic centimeter. 


Experiment 8. — Find the capacity of a test-tube, or of an irregular 
shaped cavity in any body. Weigh the body ; then fill the cavity with 
water, and weigh again. As many grams as its weight is increased, so 
many cubic centimeters is the capacity of the cavity. 

Experiment 9. — A fresh egg sinks in water. See if by dissolving 
table salt in the water it can be made to float. How does salt affect the 
density of the water ? 

Experiment 10. — Float a sensitive densimeter in water at about 
60° F. (15° C), and in other water at about 180° F. (82° C). Which water 
is denser ? 


1. Can you by placing the neck of a bottle in your mouth suck liquid 
out of the bottle ? Explain. 

2. a. What is the weight of a liter of hydrogen under a pressure of 
760 mm and at 0° C. ? h. What is the weight of a liter of dry air under 
the same conditions ? 

3. Suppose that when the barometer stands at 76 cm, the pressure of 
air within the seven-in-one apparatus (Fig. 105) is diminished one-fourth. 
The diameter of the piston being 5.75 inches, what pulling force must be 
employed to prevent the piston from being forced in, on the supposition 
that there is no friction ? 

4. Into what space must you compress 30 cu. ft. of air that its elastic 
force may be made five times as great ? 

5. If when the barometer stands at 760 mm a cubic meter of air be 
forced into a vessel whose capacity is 1000 cc, what pressure will be 
exerted upon its interior walls ? 

6. How high can naphtha be raised by a lifting pump ? 

7. Why do iron-clad vessels float in water ? 

8. A block of ice weighing 500 grams floats on water, a. What vol- 
ume of water does it displace ? h. What volume of ice is out of water ? 

9. Will ice float or sink in alcohol ? 

10. Give the density and specific density of gold, cork, and alcohol. 

11. The effective weight of a stone in water is 56 grams ; its weight in 
air is 112 grams. ' a. What is the volume of the stone ? h. What is its 
density ? 

12. How many cubic centimeters of dry air at 760 mm and 0° C. v/eigh 
as much as one cubic centimeter of water at 4° C. ? 

13. If 4 cubic feet of a body have a mass of 180 lbs. , what is its specific 
density ? 



14. How much will 1 K of copper weigh in water ? 

15. What does a piece of lead 20 X 10 X 5 cm weigh ? 

16. What would it weigh in water ? 

17. What would it weigh in mercury ? 

18. How much does a cubic foot of gold weigh? 

19. A solid body weighs 10 pounds in air and 6 pounds in water, a. 
What is the weight of an equal volume of water ? 6. What is its specific 
density ? c. What is the volume of the body ? d. What would it weigh 
if it were immersed in sulphuric acid ? 

20. A thousand-gram bottle filled with sea-water requires in addition 
to the counterpoise of the bottle 1026 grams to balance it. a. What is 
the specific density of sea-water ? h. What is the quantity of salt, etc. , 
dissolved in 1000 grams of sea-water ? 

21. A piece of cork floating on water displaces 2 pounds of water. 
What is the weight of the cork ? 

22. In which would a hydrometer sink farther, in milk or in water ? 

23. What metals will float in mercury ? 

24. a. Which has the greater specific density, water at 10° C. or water 
at 20° C. ? 6. If water at the bottom of a vessel could be raised by 
application of heat to 20° C. while the water near the upper surface had 
a temperature of 10° C, what would happen ? 

25. A block of wood weighs 550 grams ; when a certain irregular- 
shaped cavity is filled with mercury the block weighs 570 grams. What 
is the capacity of the cavity ? 

26. In which is it easier for a person to float, in fresh water or in sea- 
water ? Why ? 

27. Eig. 156 represents a beaker graduated in 
cubic centimeters. Suppose that when water stands 
in the graduate at 50 cc, a pebble stone is dropped 
into the water, and the water rises to 75 cc. a. 
What is the volume of the stone ? 5. How much 
less does the stone weigh in water than in air ? c. 
What is the weight of an equal volume of water ? 

28. If a piece of cork be floated on water in a 
graduate, and displace {i.e. cause the water to rise) 
7 cc, what is the weight of the cork ? 

29. You wish to measure out 50 g of sulphuric 
acid. To what number on a beaker graduated in 
cubic centimeters will that correspond ? 

30. State how you would measure out 80 g of nitric acid in a meas- 

Fig. 156. 


31. A measuring-beaker contains 35 cc of naphtha. What is the 
weight of the naphtha ? 

32. If 15 g of salt be dissolved in 1 liter of water without increasing the 
volume of the liquid, what will be the specific density of the solution ? 

33. A mass of lead weighs 1 K in air. What will it weigh in a 
vacuum ? 

34. A mass whose weight in air is 30 g, weighs in water 26 g, and 
in another liquid 27 g. What is the specific density of the other 
liquid ? 

35. A silver spoon weighing 150 g is supported by a string in water. 
What part of the weight is sustained by the string, and what part is sup- 
ported by the water ? 

36. Find the specific density of wax from the following data : weight 
of a given mass of wax in air is 80 g ; wax and sinker displace 102.88 cc 
of water ; sinker alone displaces 14 cc. 

37. A boat displaces 25 m^ of water. How much does it weigh ? 

38. If 50 K of stone were placed in the boat, how much water would 
it displace ? 

39. If the boat be capable of displacing 100 m^ of water, what weight 
must be placed in it to sink it ? 

40. An empty glass globe weighs 100 g ; full of air it weighs 102.4 g ; 
full of chlorine gas, it weighs 105.928 g. What is the specific density of 
chlorine gas ? 

41. What mass of alcohol can be put into a vessel whose capacity is 
1 liter ? 

42. A solid floats at a certain depth in a liquid when the vessel which 
contains it is in the air ; if the vessel be placed in a vacuum, will the 
solid sink, rise, or remain stationary ? 

43. When the volume of a body of gas diminishes, is it due to con- 
traction or compression, i.e. to internal or external forces? 

44. What is the hight of the barometer column when the atmospheric 
pressure is 10 grams per square centimeter ? 

45. A barometer in a diving-bell stands at 196 cm when a barometer 
at sea-level stands at 76 cm ; what is the depth of the surface of water 
inside the bell, below the air-exposed surface of the water above ? 

46. A measuring glass graduated in cubic centimeters contains water. 
An empty bottle floats on the water, and the surface of the water stands 
at 50 cc. If 10 g of lead shot be placed in the bottle, where will the 
surface of the water stand ? 

47. A person can lift just 200 K of copper in water ; how much can 
he lift in the air ? 


48. If a liter of gas under a pressure of 76 mm be allowed to expand 
and fill a vessel having a capacity of 10 liters, v^hat pressure will it 
exert ? 

49. A piece of lead and a piece of cork balance each other in the air. 
Which contains more matter, and how much more ? 

50. How great a buoyant force does a fluid exert on a body immersed 
in it? 




Section I. 


152. How sound originates. — Listen to a sounding church 
bell. It produces a sensation ; it is heard. The ear is the 
organ through which the sensation of hearing is produced. 
The bell is at such a distance that it cannot act directly on 
the ear ; yet something must act on the ear, and it must be 
the bell which causes that something to act. 

How does a sounding body differ from a silent body ? 

Experiment 1. — Strike a bell or a glass bell-jar, and touch the edge 
with a small ivory ball suspended by a thread ; you not only hear the 
sound, but, at the same time, you see a tremulous motion of the ball, 
caused by a motion of the bell. Touch the bell gently with a finger, and 
you feel a tremulous motion. Press the hand against the bell ; you stop 
its vibratory motion, and at that instant the sound ceases. Strike the 
prongs of a tuning-fork, and press the stem against a table ; you hear a 
sound. Thrust the ends of the prongs just beneath the surface of water ; 
the water is thrown off in a fine spray on either side of the vibrating fork. 
Watch the strings of a piano, guitar, or violin, or the tongue of a jews- 
harp, when sounding. You can see that they are in motion. 

Sound originates in niass-vibratio7i. 

How can a bell sounding at a distance affect the ear ? If 
the bell while sounding possess no peculiar property except 
motion, then it has nothing to communicate to the ear but 
motion. But motion can be communicated by one body to 
another at a distance only through some medium. 


Experiment 2. — Lay a thick tuft of cotton-wool on the plate of an 
air-pump, and on this, face downward, place a loud-ticking watch, and 
cover with the receiver. Notice that the receiver, interposed between 
the watch and your ear, greatly diminishes the sound, or interferes with 
the passage of something to the ear. Take a few strokes of the pump 
and listen ; the sound is more feeble, and continues to grow less and less 
distinct as the exhaustion progresses, until either no sound can be heard 
when the ear is placed close to the receiver, or an extremely faint one, 
as if coming from a great distance. The rehioval of air from a portion of 
the space between the watch and your ear destroys the sound. Let in 
the air again, and the sound is restored. 

The vibrations of a sonorous body cannot affect the organ of 
hearing without a continuous medium of communication between 

153. Hoiu vibratory motion, i.e. a luave, is propagated through 
an elastic medium. 

Experiment 3. — Fig. 157 represents a brass wire wound into the form 
of a spiral spring, about 12 feet long. Attach one end to a cigar-box, 
and fasten the box to a table. Hold the other end of the spiral firmly in 
one hand, and with the other hand insert a knife-blade between the turns 
of the wire, and quickly rake it for a short distance along the spiral 
toward the box, thereby crowding closer together for a little distance (B) 

Fig. 157. 

the turns of wire in front of the hand, and leaving the turns behind 
pulled wider apart (A) for about an equal distance. The crowded part 
of the spiral may be called a condensation^ and the stretched part a rare- 
faction. The condensation, followed by the rarefaction, runs with great 
velocity through the spiral, strikes the box, producing a sharp thump ; is 
reflected from the box to the hand, and from the hand again to the box, 
producing a second thump ; and by skillful manipulation three or four 
thumps will be produced in rapid succession. If a piece of twine be tied 
to some turn of the wire, it will be seen, as each wave passes it, to receive 
a slight jerking movement forward and backward in the direction of the 
length of the spiral. 


How is energy transmitted through the spring so as to 
deliver the blow on the box? Certainly not by a bodily 
movement of the spiral as a whole, as might be the case if it 
were a rigid rod. The movement of the twine shows that the 
only motion which the coil undergoes is a vibratory movement 
of its turns. Here, as in the case of water-waves, energy 
is transmitted through a medium by the transmission of 

The effect of applying force with the hand to the spiral 
spring is to produce in a certain section, B, of the spiral a 
crowding together of the turns of wire, and at A a separation ; 
but the elasticity of the spiral instantly causes B to expand, 
the effect of which is to produce a crowding together of the 
turns of wire in front of it, in the section C, and thus a for- 
ward movement of the condensation is made. At the same 
time, the expansion of B causes a filling up of the rarefaction 
at A, so that this section is restored to its normal state. This 
is not all : the folds in the section B do not stop in their 
swing when they have recovered their original position, but, 
like a pendulum, swing beyond the position of rest, thus 
producing a rarefaction at B, where immediately before there 
was a condensation. Thus a forward movement of the rare- 
faction is made, and thus a pulse or wave is transmitted with 
uniform velocity through a spiral spring or any elastic 

A wave cannot be transmitted through an inelastic soft 
iron spiral. Elasticity is essential in a medium, that it may 
transmit waves covfijposed of condensations and rarefactions; 
and the greater the elasticity, the greater the facility and 
rapidity with which a medium transmits waves. 

154. Air as a medium of luave-motion. 

Experiment 4. — Place a candle flame at the orifice a of the long tin 
tube A (Fig. 158) and strike the table a sharp blow with a book near the 



orifice b. Instantly the candle flame is quenched. The body of air in 
the tube serves as a medium for transmission of motion to the candle. 

Fig. 158. 

Is the motion transferred that of a current of air through the tube (a 
miniature wind), or is it a vibratory motion ? Burn touch-paper i at the 
orifice 6, so as to fill this end of the tube with smoke, and repeat the last 

Evidently, if the body of the air be moved along through 
the tube, the smoke will be carried along with it. The candle 
is blown out as before, but no smoke issues from the orifice a. 
It is clear that there is no translation of material particles 
from one end to the other, — nothing like the flight of a rifle 
bullet. The candle flame is struck by something like a ^^ulse 
of air, not by a luind} 

Air is a fluid, and has therefore only volume elasticity. 
The only waves it can propagate are waves composed of com- 
pressions and rarefactions. In a previous chapter we have 
seen how a wave is the result of a transmission of harmonic 
motion or harmonic vibrations through a series of particles. A 
sound-wave consists of a succession of particles of the sound 
medium vibrating harmonically and successively and in the 

1 To prepare touch-paper, dissolve about a teaspoonful of saltpetre in a half -tea- 
cupful of hot water, dip unsized paper in the solution, and then allow it to dry. The 
paper produces much smoke in burning, but no flame. 

2 If a membrane be tied tightly over the orifice h and a sudden blow be given it 
{e.g. by snapping it with a finger), the vibratory character of the motion communi- 
cated through the tube is well shown by the flame being first driven from the orifice a 
and immediately afterward drawn toward it. 


same direction as that in which the sound-wave moves. There 
are two important distinctions between these waves and waves 
of water, or waves sent along a cord when one end is shaken : 
the former consist of condensations and rarefactions ; the 
latter of elevations and depressions. In the former, the 
vibration of the particles is "in the same line Avith the path of 
the wave, and hence they are called longitudinal vibrations ; 
in the latter the vibrations take place in planes at some angle 
to the path of the wave, and are therefore called transverse 

Boys often amuse themselves by inflating paper bags, and 
with a quick blow bursting them, producing with each a single 
loud report. First the air is suddenly and greatly condensed 
by the blow, and the bag is burst ; the air now, as suddenly 
and with equal force, -expands, and by its expansion condenses 
the air for a certain distance all around it, leaving a rare- 
faction where just before had been a condensation. If many 
bags were burst at the same spot in rapid succession, the 
result would be that alternating shells of condensation and 
rarefaction would be thrown off, all having a common center, 
enlarging as they advance, as do the ripples formed by stones 
dropped into water ; except that, in this case, the waves are 
not like rings, but hollow globes ; not circular, but spherical. 
In this manner sound-waves produced by the vibration of a 
sounding body travel through the air. 

As a wave advances, each individual air-particle concerned 
in its transmission performs a short excursion to and fro in 
the direction of a straight line radiating from the center of 
the shells or hollow globes. A sound-ivave travels its oivn 
length in the time that a jparticle occupies in going through one 
complete vibration so as to he ready to start again. 

Experiment 5. — Take a strip of black cardboard 4.5 inches X 1 inch. 
Cut a slit about one-sixteenth of an inch wide lengthwise and centrally 
through the strip nearly from end to end. Place the slit across the page 



just below Fig. 159, and draw the book along underneath in the dh-ection 
of the arrow. Imagine that the short dark dashes seen through the slit 
represent a series of air-particles, and the slit itself represents the direc- 
tion in which a series of sound-waves is travelling. It will be seen that 

Fig. 159. 

each air-particle moves a little to and fro in the direction m which the 
sound-waves travel, and comes back to its starting-point ; but the conden- 
sations and rarefactions, represented by a group (half a wave-length) of 
dots becoming alternately closer together or farther apart, are transmitted 
through the whole series of air-particles. 

155. Nature of sound and sound-ivaves. — Sound is a sen- 
sation caused usually hy air-waves heating upoii the organ of 

1 As commonly used tlie term sound is ambiguous, being applied to both a 
sensation and the physical cause of the sensation. In a scientific treatise ambiguity 
and consequent confusion are disastrous. No apology, therefore, is required for 
restricting the term to its legitimate signification. With sound itself^'e have little 


Sound-waves are waves in any mediuni (usually air) that are 
capable of producing the sensation of sound. A body vibrating 
in an elastic medium, e.g. in air, does not necessarily produce 
sound-waves ; in other words not all waves are sound-waves. 
For example, the energy of the vibrations may be insufficient, 
or the vibrating body may be so small (or the medium so 
rare) that it cuts through the medium without condensing it 
sufficiently to produce audible effects. 

156. Solids and liquids are media capable of transmitting 

Experiment 6. — Lay a watch, with its back downward, on a long 
hoard (or table), near to one of its ends, and cover the watch with loose 
folds of cloth until its ticking cannot be heard through the air in any 
direction at a distance equal to the length of the board. Now place the 
ear in contact with the farther end of the board, and you will hear the 
ticking of the watch very distinctly. 

Experiment 7. — Place one end of a long pole on a cigar box, and 
apply the stem of a vibrating tuning-fork to the other end; the sound- 
vibrations will be transmitted through the pole to the box, and a sound 
will be given out by the box, as though that, and not the tuning-fork, 
were the origin of the sound. 

Experiment 8. — Place the ear to the earth, and listen to the rumbling 
of a distant carriage ; or put the ear to one end of a long stick of timber, 
and let some one gently scratch the other end with a pin. 

Section II. 


157. Speed of sound-waves dependent on elasticity and den- 
sity of mediuin. — It may be demonstrated -^ that in simple 
harmonic motion, the velocity with which a particle of an elastic 

to do, as this is a physiological rather than a physical phenomenon. No more 
appropriate name than sound-wave can be applied to the physical agent with which 
we are to deal ; it stiggests at once the reality, and is not suggestive of some vague 
mysterious thing shot through space. 
1 See Barker's Physics, p. 219. 


medium vibrates, and therefore the speed of propagation in the 
TYiedium (Jience, the speed of a sound-wave^, is directly propor- 
tional to the square root of the elasticity of the medium, and 
inversely proportional to the square root of its density. The 
relation of these quantities is shown in the formula 

If the elasticity and density of the medium vary alike, and 
in the same direction, it is evident that the speed of the 
sound-wave is unaffected. Hence the speed of a sound-wave 
is unaffected by barometric hight, or elevation above sea-level. 
Temperature, however, affects only the density of air. Ele- 
vation of temperature of the air diminishes the density of the 
air, and therefore tends to increase the speed of the sound- 
wave. Moisture in the air renders it less dense (pressure 
remaining constant), and thereby tends to increase the speed. 
The velocity of a sound-wave is greatest in the direction of 
the wind. Speed of sound-waves is very nearly independent 
of pitch and intensity. 

The greater density of solids and liquids, as compared with 
gases, tends, of course, to diminish the speed of sound-waves; 
but their greater incompressibility more than compensates 
for the decrease of speed occasioned by the increase of den- 
sity. As a general rule, solids are more incompressible than 
liquids ; hence sound-waves generally travel faster in the 
former than in the latter. For example, sound-waves travel 
in water about four times as fast as in air, and in iron and 
glass sixteen times as fast. 

The speed of sound-waves in free air at 0° C. is 332.4 m 
(nearly 1091 ft.) i^er second. The increase of speed per 
degree C. is .608 m (23.9 in.). The speed in other gases = 


—j^, in which v is the speed in air and d the density of the 
given gas referred to air. For example, in hydrogen, whose 


density is ^ that of air, the speed at 0° C. is about 4163 ft. 
per second. The speed of sound-waves in any medium may 
be calculated from the formula given above and by experi- 
mental methods to be given further on. 

Section III. 


158. Energy of sound-ivaves depe7ids on the amplitude of 
vibration. — Gently tap the prongs of a tuning-fork and dip 
them into water, — the water is scarcely moved by them ; 
increase the energy of the blow, — the vibrations become 
wider, and the water spray is thrown with greater force and 
to a greater distance. The same thing occurs when the fork 
vibrates in the air ; though we do not see the air-particles as 
they are batted by the moving fork, yet we feel the effects 
as a sound sensation, and we judge of their energy by the 
intensity of the sensation which they produce. 

Fix your attention upon a particle of air as a sound-wave 
passes it. A harmonic motion is. impressed upon it. At a 
certain point of its excursion its velocity is at its maximum. 
Now since the energy of a moving particle varies as the 
square of its velocity, the intensity of the impact which it is 
capable of producing upon the tympanum of the ear is propor- 
tional to the square of this maximum velocity. 

It is also clear that if the amplitude of vibration of a 
particle be doubled while its period remains constant, its 
velocity is doubled, and therefore its energy is increased four- 
fold. Hence, (1) measured mechanically, the energy of a 
sound-wave is proportional to the square of the amplitude of the 
vibration of particles, or, it is proportional to the square of the 
maximum velocity of the vibrating particles. An amplitude of 
less than to-q oV"o o o ^"^ ^^ sufficient to cause hearing. 


Loudness of sound refers to the intensity of a sensation. 
We have no standard of measurement for a sensation, so we 
are compelled to measure the energy of the sound-wave, 
knowing at the same time that loudness is not projjortional to 
this energy. 

159. Energy of sound-ivaves depends iipjon the density of 
the niedmm. — In the experiment with the watch under the 
receiver of the air-pump (p. 187), the sound grew feebler as 
the air became rarer. Aeronauts are obliged to exert them- 
selves more to make their conversation heard when they 
reach great hights than when in the denser lower air. In 
diving-bells persons are obliged to speak in undertones. In a 
rare medium a vibrating body during a single vibration either 
sets in motion fewer particles, as in the case of the partially 
exhausted receiver, or, as in the case of hydrogen gas, it sets 
in motion particles of less mass than in a dense medium ; 
consequently it parts with its energy more slowly, and the 
sound is consequently weaker. 

(2) The energy of gaseous sound-waves increases with the 
density of the niediimi in which they are produced. 

160. Energy of sound-waves depends on distance from their 
source. — It is a matter of every-day observation that the 
loudness of a sound diminishes very rapidly as the distance 
from the source of the waves to the ear increases. As a 
sound-wave advances in an ever-widening sphere, a given 
quantity of energy becomes distributed over an ever-increasing 
surface ; and as a greater number of particles partake of the 
motion, the individual particles receive proportionately less 
energy ; hence it follows, — as a consequence of the geomet- 
rical truth, that " the surface of a sphere varies as the square 
of its radius," — that (3) the energy of a sound-wave varies 
inversely as the square of the distance from the source. This is 
known as the Law of Inverse Squares.^ For example, if two 

1 That the Law of Inverse Squares is applicable to sound-waves is sometimes 


persons, A and B, be respectively 500 and 1000 rods from a 
gun when it is discharged, the waves that reach A will have 
four times the energy that the same waves have when they 
reach B. 

161. Speaking-tubes. 

Experiment 9. — Place a watch at one end of the long tin tube (Fig. 
158), and the ear at the other end. The ticking sounds very loud, as 
though the watch were close to the ear. 

Long tin tubes, called speaking-tubes, passing through many 
apartments in a building, enable persons at the distant ex- 
tremities to carry on conversation in a low tone of voice, 
while persons in the various rooms through which the tube 
passes hear little or nothing. The reason is that the sound- 
waves which enter the tube are prevented from expanding, 
consequently the energy of the sound-waves is not affected by 
distance, except as it is wasted by friction of the air against 
the sides of the tube, and by internal friction due to the 
viscosity of the air. 

162. Energy of soimd-iuaves depends on the homogeneousness 
of the transmitting medium. — Observations and experiments of 
Humboldt, Tyndall, and Henry have established the following 
facts : Eain, hail, snow, and fog offer little or no obstruction 
to the passage of a sound-wave. The air associated with a 
fog is, as a general rule, highly homogeneous and favorable 
to the transmission of sound. An atmosphere optically opaque 
may be acoustically transparent, and vice versa; hence the 
great value of fog horns. Streams of air differently heated, 
or saturated in different degrees with aqueous vapor, though 
invisible to the eye, form acoustic clouds which may greatly 
interfere with the propagation of sound-waves. 

163. Energy of sound-ivaves affected by winds. — It not 
infrequently happens that sound-waves are audible two or 

called in question. For an experimental demonstration of its applicability, see 
" Contributions from the Physical Laboratory, Mass. Inst. Technology, 1876." 



three times as far to the leeward as to the windward. Sound- 
waves are borne along with the wind, but, of course, are 
impeded by it when the directions of their motions are opposed 
to it. 

Section IV. 


164. Beflection. — So long as sound-waves are not ob- 
structed in their motion they are propagated in the form of 
concentric spheres ; but when they meet with an obstacle, 
they follow the general law of elastic bodies; that is, they 
return upon themselves, forming new concentric waves, called 
reflected waves, which seem to emanate from a second center 
on the other side of the reflecting body. This phenomenon 
is called the reflection of soimd-iuaves. A (Fig. 160) repre- 
sents a vibrating particle or a sonorous center from which 

Fig. 160. 

emanates a series of waves. P Q represents an obstacle with 
a flat surface turned toward the waves. Take, for example, 
the incident wave MCDN, emitted from the center A; the 
corresponding reflected wave is represented by the arc C K D 


of a circle whose center a is as far beyond the obstacle P Q 
as A is in front of it. 

Join any point, C, of the reflecting surface to the sonorous 
center, and the line A C represents one of an infinite number 
of directions in which energy is transmitted by a sound-wave. 
Such a line may conveniently be called a sound-ray. Let fall 
the line H C normal to the surface at the point of incidence C. 
The angle A C H is called the angle of incidence. The ray 
A C after reflection takes the direction C B, which is a prolon- 
gation of «C. The angle BCH is called the angle of reflection. 
An observer at B receives sound-waves not only directly from 
A in the line AB, but also from C in the line CB. Hence he 
hears two sounds, one (to speak in common parlance) proceed- 
ing from point A, and the other from point C. The latter 
travels from A to C and from C to B, a longer distance than 
AB, and is therefore heard later than the former. If the 
interval of time between their arrivals at B be greater than 
about a fifth of a second, the ear is able to separate the two 
sensations and the latter appears as an eclio. If the interval 
of time be too short, then only a single and perhaps somewhat 
blurred and indistinct sound is heard. The latter phenomenon 
is usually called resonance. Such an effect is experienced 
frequently by a person listening to his own voice in a large 

If the obstacle PQ present a concave surface, the wave- 
front after reflection will be less convex, and may become 
plane or even concave according to the degree of the concavity 
of the reflector and the position of the sounding body. 

165. Sound-waves reflected by concave mirrors. 

Experiment. — Place a watch at the focus A (Fig. 161) of a concave 
mirror G. At the focus B of another concave mirror H, place the large 
opening of a small tunnel, and with a rubber connector attach the bent 
glass tube C to the nose of the tunnel. The extremity D being placed 
in the ear, the ticking of the watch can be heard very distinctly, as though 


it were somewhere near the mirror H. Though the mirrors be 12 feet 
apart, the sound will be louder at B than at an intermediate point E. 

How is this explained ? Every air-particle in a certain 

radial line, as A.c, receives and transmits motion in the 

direction of this line ; the last particle strikes the mirror 

at c, and bein^ perfectly , ^7 

G ^jC- ^ H 

elastic, bonnds off in the AT 



V .If y 


Fig. 161. 

direction c c', commnnicat- 
ing its motion to the par- 
ticles in this line. At c' 
a similar reflection gives 
motion to the air particles 
in the line c'B. In consequence of these two reflections, all 
divergent sound-rays, as Ad, Ae, etc., that meet the mirror G, 
are there rendered parallel, and afterwards rendered conver- 
gent at the mirror H. The practical result of the concentra- 
tion of this scattering energy is, that a sound of great intensity 
is heard at B. The points A and B are called the foci of the 
mirrors. The front of the wave as it leaves A is convex, in 
passing from G to H it is plane, and from H to B concave. 
If you fill a large circular tin basin with water, and strike 
one edge with a knuckle, circular waves with concave fronts 
will close in on the center, heaping up the water at that point. 

Long "whispering-galleries" have been constructed on this 
principle. Persons stationed at the foci of the concave ends 
of the long gallery can carry on a conversation in a whisper 
which persons between cannot hear. The external ear is a 
wave-condenser. The hand held concave behind the ear, by 
its increa'sed surface, adds to its efficiency. 

166. Refraction. — If you place your ear at the small end 
of a tunnel, C (Fig. 162), and listen to the ticking of a watch, 
A, about 4 meters distant, and then introduce a collodion 
balloon, B, filled with carbonic acid gas between your ear 



and the watch, and very near the latter, the sound becomes 

The cause is obvious : for let the curved lines a, h, c, etc., 
represent sections of sound-waves with convex fronts, and B 

Fig. 162. 

a spherical body of carbonic acid gas which is denser than air ; 
then it is clear that, owing to the slower progress of the waves 
in the denser gas, they would become flattened on entering 
this gas, and the waves of convex fronts may be changed to 
waves of plane fronts. Again, points at the extremities of 
the waves, having less distance to travel in the denser gas 
than points near the center, would emerge first and get in 
advance, and thus the wave fronts which are plane or nearly 
so while wholly in the dense gas, become concave on leaving 
it. By these changes in the form of the wave fronts, sound 
energy which was originally becoming diffused through wider 
and wider space, and therefore becoming less intense as it 
progressed, is so changed in direction in passing into and out 
of a medium of greater density, that the energy is finally 
concentrated at a distant point, as at C, and thereby inten- 

Any change in direction of sound, caused by passing from 
a medium of a certain density into a medium of different 
density, is called refraction. 

167. Diffraction. — When sound-waves encounter an ob- 
stacle, a series of secondary waves are formed with the edges 



of tlie obstacle as centers. These waves appear to flow around 
behind the object, so that the obstacle is able to produce only 
a partial sound-shadow. ^'Sounds heard around a corner" 
are thus accounted for. This bending, as it were, of waves 
around an obstacle is called diffraction. 

Section V. 


168. Reenforceynent of sound-waves. 

Experiment 1. — Set a diapason in vibration ; you can scarcely hear 
the sound unless it is held near the ear. Press the stem against a table ; 
the sound rings out loud, but the waves seem to proceed from the table. 

When only the fork vibrates, the prongs, presenting little 
surface, cut their way through the air, producing very slight 
condensations, and consequently waves of little intensity. 
When the fork rests upon the table, the vibrations are com- 
municated to the table ; the table with its larger surface 
throws a larger mass of air 
into vibration, and thus 
greatly intensifies the sound- 
waves. The strings of the 
piano, guitar, and violin owe 
as much of their loudness of 
sound to their elastic sound- 
ing-boards as the fork does 
to the table. 

169. Reenforcenient hy bod- 
ies of air ; resonators. 



— e, 

a — :.'::: 


Fig. 163. 

Experiment 2. — Take a glass 
tube, A (Fig. 163), 16 inches long 
and 2 inches in diameter ; thrust one end into a vessel of water, C, and 
hold over the other end a vibrating diapason, B, that makes (say) 256 


vibrations in a second. Gradually lower the tube into the water, and 
when it reaches a certain depth, i. e. when the column of air o c attains a 
certain length, the sound becomes very loud ; as the tube is lowered 
below this point, the sound rapidly dies away. 

Columns of air, as well as sounding-boards, serve to reenforce 
sound-waves. The instruments which enclose the columns of 
air are called resonators. Unlike sounding-boards, they can 
respond loudly to only one tone, or to a few tones of widely 
different pitch. 

How is this reenforcement effected? When the prong a 
moves from one extremity of its arc a' to the other a", it 
sends a condensation down the tube ; this condensation, 
striking the surface of the water, is reflected by it up the 
tube. Now suppose that the front of this reflected conden- 
sation should just reach the prong at the instant it is starting 
on its retreat from a^^ to a' ; then the reflected condensation 
will conspire with the condensation formed by the prong in 
its retreat to make a greater condensation in the air outside 
the tube. Again, the retreat of the prong from a^^ to <x' 
produces in its rear a rarefaction, which also runs down the 
tube, is reflected, and will reach the prong at the instant it is 
about to return from ct' to a", and to cause a rarefaction in its 
rear ; these two rarefactions moving in the same direction 
conspire to produce an intensified rarefaction. The original 
sound-waves thus combine with the reflected, to produce 
resonance 5 but this can happen only when the like parts of 
each wave coincide each with each ; for if the tube were 
somewhat longer or shorter than it is, it is plain that conden- 
sations and rarefactions would meet in the tube, and tend to 
destroy each other. 

The loudness of sound of all wind instruments is due to 
the resonance of the air contained within them. A simple 
vibratory movement at the mouth or orifice of the instrument, 
scarcely audible in itself (such as the vibration of a reed in 



reed pipes, or a pulsatory movement of the air, produced by 
the passage of a thin sheet of air over a sharp wooden or 
metallic edge, as in organ pipes, flutes, and flageolets, or more 
simply still by the friction of a gentle stream of breath from 
the lips sent obliquely across the open end of a closed tube 
or pen-case), is sufficient to set the large body of enclosed air 
in the instrument into vibration, and the sound thus reenforced 
becomes audible at long distances. 

Experiment 3. — Attach a rose gas-burner, A (Fig. 164), to a metal 
gas-tube about 1 m in length, and connect this by a 
rubber tube with a gas-nipple. Light the gas at the 
rose burner, and you will hear a low, rustling noise. 
Remove the conical cap from the long tin tube (Fig. 
158), support the tube in a vertical position, and gradu- 
ally raise the burner into the tube ; when it reaches a 
certain point not far up, the body of air in the tube will 
catch up the vibrations, and give out deafening sound- 
waves that will shake the walls and furniture in the 

170. Measuring ivave-lengths and the speed 
of sound-iuaves. — Experiments like that de- 
scribed on p. 201 enable us readily to meas- 
ure the length of the wave produced by a fork 
whose vibration number is known, and also to 
measure the velocity of sound-waves. It is 
evident that if a condensation generated by 
the prong of the fork in its forward movement from «' to «" 
(Fig. 163) meet with no obstacle, its front, meantime, will 
traverse the distance od, or twice the distance oc ; hence the 
length of the condensation is the distance od. But a conden- 
sation is only one-half of a wave, and the passage of the 
prong from a' to a" is only one-half of a vibration; conse- 
quently the distance oc? is one-half of a wave-length, and the 
distance o c is one-fourth of a wave-length. The measured dis- 
tance of oc in this case is about 13.13 inches ; hence the 

Fig. 164. 



length of wave produced by a C'-fork making 256 vibrations 

in a second is (13.13 inches X 4 =) 52.5 inches = 4.38 feet. 

And since a wave from this fork travels 4.38 feet in -^^^ of a 

second, it will travel in an entire second (4.38 feet X 256 =) 

1121 feet. The distance oc varies with the temperature of 

the air. 

It is evident that the three quantities expressed in the 



wave-length = — rr — -. 

number oi vibrations 

bear such a relation to one another that if any two be known, 
the remaining quantity can be computed. It will further be 
observed that ivith a given velocity the ivave-length varies 
inversely as the number of vibrations ; i.e. the greater the num- 
ber of vibrations per second, the shorter the wave-length. 
171. Interference of soimd-ivaves. 

Experiment 4- 

Hold a vibrating diapason over a resonance-jar, as in 
Fig. 165. Koll the diapason over 
slowly in the fingers. At certain 
points a quarter of a revolution 
apart, when the diapason is in an 
oblique position with reference to 
the edge of the jar as represented 
in the figure, the reenforcement 
from the tube almost entirely dis- 
appears, but it reappears at the in- 
termediate points. That is, there 
are four intervals in the space 
around the fork where the two 
series of waves generated by the 
two tines interfere to produce 
mutual destruction. These are' 
called technically the cones of silence. Return to the position where 
there is no resonance, and enclose in a loose roll of paper the prong 
farthest from the tube, without touching the diapason, so as to prevent 
the sound-waves produced by that prong from passing into the tube; the 



resonance resulting from the vibrations of the other prong immediately 

^Experiment 5. — Select two of the tubes (Fig. 188) of nearly the same 
length, blow through them, and notice the peculiar throbbing sound 
produced by the interference of the two sounds. 

Experiment 6. — Stop one of the orifices of a bicyclist's 
whistle (Fig. 166), and sound one whistle at a time. The 
sound of each is clear and smooth. Sound both whistles at 
the same time, and you obtain the usual rough and discord- 
ant sound. 

The two whistles of unequal length give out waves of 
slightly different length, so that at certain short intervals 
both waves will interfere in the same phase (i. e. condensation 
with condensation) and produce intensified sounds which 
are heard at long distances, while at other intervals they 
interfere in opposite phases (i.e. condensation with rarefac- 
tion), and the result of their mutual destruction is to cause 
the otherwise smooth sound to become broken or rattling. 

Two sound-waves may combine to produce a sound louder 

Fig. 166. 

or weaker than either alone would produce^ or may even cause 

silence. This combination of sound-waves to produce a louder or weaker 

sound is called interference. 

172. Forced and sympathetic vibrations. 

Experiment 7. — Suspend from a frame several pendulums. A, B, C, 
etc. (Fig. 167). A and D are each 3 feet long, 
C is a little longer, and B and E are shorter. 
Set A in vibration, and slight impulses will be 
communicated through the frame to D, and 
cause it to vibrate. The vibration-period of D 
being the same as that of A, all the impulses 
tend to accumulate motion in D, so that it soon 
vibrates through arcs as large as those of A. 
On the other hand, C, B, and E, having differ- 
ent rates of vibration from that of A, will at 
first acquire a slight motion, but soon their 
vibrations will be in opposition to those of A, 

and then the impulses received from A will tend to destroy the slight 

motion they had previously acquired. 

Experiment 8. — Press down gently one of the keys of a piano so as to 



Fig. 168. 

raise the damper without making any sound, and then sing loudly into 
the instrument the corresponding note. The string corresponding to this 
note will be thrown into vibrations that can be heard for several seconds 
after the voice ceases. If another note be sung, this string will respond 
only feebly. 

Eaise the dampers from all the strings of the piano by pressing the 
foot on the right-hand pedal, and sing strongly some note into the piano. 
Although all the strings are free to vibrate, only those will respond loudly 
that correspond to the note you sing, i.e. those that are capable of making 
the same number of vibrations per second as are produced by your voice. 
Experiment 9. — Take two forks, A and B (Fig. 168), tuned exactly in 

unison, and mounted on reso- 
nance-boxes, and place them 
from three to ten meters apart. 
Fasten, by a bit of sealing-wax, 
a thread to a thin piece of glass 
12 mm square (glass used for 
microscopic mountings is the 
best, or a piece of photographic 
tintype plate will answer well), 
and suspend so as to touch a 
corner of one of the prongs of the fork B. Set the fork A in vibration 
by drawing a resined bass-viol bow strongly across the ends of its prongs. 
In about ten seconds stop the vibrations of A with the fingers, and you 
will see and hear the piece of glass rattling against the prong of the fork 
B ; remove the glass, and place the ear near the fork B, or better, the 
open end of the box, and you may hear a distinct sound, showing that 
the fork B has been thrown into a state of vibration by the fork A. 

So the pulses that traverse the air between the forks, so 
gentle that only the sensitive organ of the ear can perceive 
them, become great enough to move the rigid steel when the 
energy of their blows, dealt at the rate of perhaps 512 in a 
second, accumulates. The large number of blows makes up 
for the feebleness of the individual blows. 

These experiments show that a vibrating body tends to 
make other bodies near it vibrate, even if their periods of 
vibrations be different. Vibrations of this kind, such, for 
example, as those of B, C, and E in Exp. 7 and those 



generated in the sounding-boards of pianos, violins, etc., are 
called forced vibrations. But if the period of the incident 
waves of air be the same as that of the body which they 
cause to vibrate, the amplitude and intensity of the vibrations 
become very great, like that of the pendulum D, and those of 
the piano strings which gave forth the loud sounds. Such 
are called syvipathetic vibrations. 

Section VI. 


173. On ivhat pitch depends. 

Exjperiment 1. — Draw tlie finger-nail or a card slowly, and then 
rapidly, across tlie teetli of a comb. The two sounds produced are com- 
monly described as low or grave, and high or acute. The hight of a 
musical sound is its pitch. 

Experiment 2. — Cause the circular 
sheet-iron disk A (Fig. 169) to rotate, 
and hold a corner of a visiting-card so 
that at each hole an audible tap shall be 
made. Notice that when the separate 
taps or noises cease to be distinguishable, 
the sound becomes musical ; also, that 
the pitch of the musical sound depends 
upon the rapidity of the rotation, i.e. 
upon the frequency of the taps. 

Experiment 3. — Hold the orifice of a 
tube B so as to blow through the holes 
as they pass. When rotating slowly, 
separate puffs, from which it hardly 
seems possible to construct a musical 
sound, are heard. When, however, the 
ear is no longer able to detect the sepa- 
rate puffs, the sound becomes quite musi- 
cal, and the pitch rises and falls with the 

Fig. 169. 

Pitch depends upon the number of sound-waves striking the 


ear per second. If the source of the sound-waves and the receiv- 
ing ear he both stationary, the pitch depends upon the frequency 
of vibration, or wave-length; i.e. the greater the number of 
vibrations per second, or the shorter the wave-length, the higher 
the pitch. 

Since pitch depends upon the number of sound-waves strik- 
ing the ear per second, a sound must rise in pitch if we 
rapidly approach the source of the sound-waves, or the source 
rapidly approach us, as evidently more sound-waves will then 
strike the ear per second than otherwise would happen. The 
pitch of the whistle rises on the rapid approach of a locomo- 
tive, and falls again as the engine travels away.^ 

1 It may be of interest to consider more in detail two cases of relative motion 
between the ear and the sounding body : 

Case I. Source stationary and ear approaching it with velocity v^ (Fig. 170) per 


Fig. 170. 

second. Let v = the velocity of the sound-waves per second, and n = the number of 
vibrations made by the vibrating body {e.g. a fork) per second. The wave-length is 

- and the number of waves in the space v-y is t\ -f- - or — *- • The number of waves 


striking the ear per second is, therefore, n + — ■— , or w I 1 + ^1. If Vi= v, the 

perceived is an octave above the normal pitch of the fork. If v^ = r, but the motion 
of the ear be away from the fork, this quantity becomes n (1—1), or zero, and no 
sound-waves reach the ear. 

Case II. Ear stationary, but the fork moves towards it with a velocity of v^ per 
second. While the fork moves from A to B (Fig. 171), the first sound-wave sent out 

from the fork at the beginning of the second has travelled all the way from A to C. 
The last wave sent out in that second of time is just starting from B when the very 
first wave has reached C. Thus there are n waves between B and C, and the length 


174. Hoiv to find the vibration-frequency of a tone. — The 
siren. — The perforated wheel described above is a cheap 
imitation of a portion of an important instrument called a 
siren. The instrument complete has an attachment called a 
counter^ which shows the number of revolutions the wheel 
makes in a given time. 

Suppose that it is required to ascertain the number of 
vibrations per second necessary to produce a given pitch. 
Take some instrument that gives the required pitch, e.g. a 
tuning-fork, and set it in vibration ; also rotate the siren, 
causing the pitch of its sound gradually to rise until it corre- 
sponds with the pitch of the fork ; then, sustaining that 
pitch, set the counter in operation, and at the end of a given 
time read off the number of revolutions made by the wheel ; 
this number multiplied by the number of holes in the wheel 
gives the number of sound-waves produced by the wheel 
during the given time, and the number of vibrations made by 
the fork in the same time ; and this number divided by the 
number of seconds employed gives the number of vibrations 
that must be made in a second by any instrument in order to 
produce a sound of the same pitch. With the siren we may 
even determine the number of vibrations made by the wing of 
a fly which buzzes around our ears. 

The vibration frequency of a fork may be easily found by means 
of an apparatus called a vibrograph. One of the tines of the fork a 
(Fig. 172) has a small elastic indicator attached to its extremity. 
The sharp point of this indicator touches a smoked glass plate, k, 
below. Above the glass plate is suspended a pendulum with a 
heavy bob. Beneath the bob is another indicator which just grazes 

of each is only ^' and the number of vibrations received by the ear per second is 

V -^ ^' or — This shows that when Vn= v, this fraction has the value oo ; 

n v — Vz 

also that in order to get the octave above the normal, v^ must equal -• 

It is obvious that in Case 11. there is a shortening of the waves, but there is no 
shortening in Case I. 



the glass as it passes the lower part of its arc. The experimenter 
first finds the exact fraction of a second occupied by the pendulum 
in making one complete or double vibration. The fork is then put 
in vibration and the block h carrying the glass plate is drawn along 

Fig. 172. 

beneath the style, which marks upon the glass a wave line. Imme- 
diately after the glass is put in motion the pendulum is set swinging 
and allowed to traverse the plate width- wise three times, making, 
with its indicator, three lines athwart the wave line. Now the 
interv^ of time between the instants when the first and the third 
of these lines are made is the time of one complete vibration. The 
number of vibrations which the fork made in this interval may be 
determined from the sinuous curved line intervening between the 
lines made by the pendulum. The number of vibrations made by the 
fork in a certain fraction of a second having been ascertained in this 
manner, the vibration number per second is calculated therefrom. 

175. Distmction between 7ioise and 'musical sound. — If the 
body that strikes the air deal it but a single blow, like the 
discharge of a fire-cracker, the ear receives but a single shock, 
and the result is called a noise. If several shocks be slowly 
received by the ear in succession, the ear distinguishes them 
as so many separate noises. If, however, the body that strikes 
the air be in vibration, and deal it a great number of little 
blows in a second, or if a large number of fire-crackers be 


discharged one after another very rapidly, so that the ear is 
unable to distinguish the individual shocks, the effect produced 
is that of one continuous sound, which may be pleasing to the 
ear ; and, if so, it is called a musical sound. But continuity 
of sound does not necessarily render it musical. The sound 
produced by a hundred children beating various articles in a 
room with clubs might not be lacking in continuity, but it 
would be an intolerable noise. There would be wanting those 
elements that please the ear ; viz. regularity both in perio- 
dicity and intensity of the shocks which it receives. The 
distinction between music and noise is, generally speaking, 
a distinction between the agreeable and the disagreeable, 
between regularity and confusion. 

176. Musical scale. — Suppose a body, e.g. a tuning fork, 
to make 261 vibrations per second, the sound produced is 

recognized by our musical sense as the note ^ T^ which 

corresponds with the so-called middle C (c', or French utg) of 
a piano tuned to the national standard pitch. ^ 

The pitch of a sound produced by twice as many vibrations 
as that of another sound is called the octave of the latter. 
Between two such sounds the voice rises or falls, in a manner 
very pleasing to the ear, by a definite number of steps called 
micsical intervals. This gives rise to the so-called diatonic 
scale, or gamut. Long before any one had attempted to find 
the frequency of vibration of a sounding body, men had used 
a succession of sounds, differing in pitch, determined only by 
their musical sense and not by arbitrary agreement. The 
number of vibrations which shall constitute a given note is 
purely arbitrary, and differs slightly in different countries ; 

1 In a conTention of piano manufacturers held in New York it was decided that 
the national pitch to go into effect July 1, 1892, should he the standard French, 
Austrian, and Italian pitch of 435 (Ag) double vibrations in a second at 68° F. 


but the ratios between the vibration numbers of the several 
notes of the gamut and the vibration number of the first or 
fundamental note of the gamut, are the same among all 
enlightened nations. 

The successive tones of the diatonic scale of C are related 
to one another Vv^ith respect to vibration frequency as follows : 


-&- — ^- 


























: 288 

: 320 : 


: 384 : 


: 480 : 



: f 

: 1 : 


: 1 ' 


: -V- : 


No. of vi- 



The ear is wholly incapable of determining the number of 
vibrations corresponding to a given tone, but it is capable of 
determining with wondrous precision the ratio of the vibration 
numbers of two notes ; hence all music must depend upon the 
recognition of such ratios, and for this reason the vibration 
ratios given above are of the utmost importance. An octave 
below c' is c ; two octaves below, Ci, and so on. In a similar 
manner the octaves below any other tone are indicated. 

The following are some of the various musical intervals occurring 
w^ithin the diatonic scale : Minor second^ e' : f^ or V : of' : : 15 : 16 ; 
Major second, of : d', f : g', or a' : b' : : 8 : 9 ; Minor third, e' : g' or 
a' : c^' : : 5 : 6 ; Major third, of : e', f ' : a', or g' : b' : : 4 : 5 ; fourth, 
of \l', d' : g', e' : a', or g' : c'' : : 3 : 4 ; jifth, of : g', e' : b', or f : Q," : : 
2 : 3, etc. 

177. Limits of scale and of audibility. — The lowest note 
of a 7-J octave piano makes about 27-|- vibrations per second; 
the highest, about 4,224 vibrations per second ; but these 
extreme notes have little musical value, and the lowest notes 
are chiefly used for their harmonics only (see p. 220). 
The range of the human voice lies between 61 and 1305 


vibrations per second/ or a little more than three octaves ; an 
ordinary singer has about the compass of two octaves. 

The ear is capable of hearing vibrations far exceeding in 
number the requirements of music. It can appreciate sounds 
arising from 32 to 38,000 vibrations ^ per second, i.e. a range 
of about eleven octaves, and a corresponding range of wave- 
length between seventy feet and three or four tenths of an 
inch. These numbers vary considerably, however, with the 
person. Exceptional ears can hear as many as 50,000 vibra- 
tions. Some ears can hear a bat's cry, or the creaking of a 
cricket; others cannot. Singing mice are sometimes placed 
on exhibition. Of those who go to hear them, some can hear 
nothing, others a little, and others again can hear much. In 
the ability to hear sharp sounds, no animal is superior to the 
cat, which finds her prey in the dark by its squealing. High 
tones are heard with difficulty in the presence of low ones. 
A lower tone tends to drown a higher one. 


1. Find the vibration number for each note of the scale of which c'' is 
the first note. 

2. What is the vibration number of c an octave below c' ? 

3. Find the wave-length corresponding to each note of the scale of 
which c' is the first, when the temperature of the air is 16° C. ? 

4. Find the length of a resonance tube (disregarding its diameter) 
closed at one end, which will respond to c" when the temperature is 
16° C. ? 

5. a. The interval between e' and c'' is called a minor sixth ; what is 
the vibration ratio for this interval ? 6. What is the note a minor sixth 
above a' ? 

6. Make out a series of fractions which shall express the vibration 
ratios of each tone in the diatonic scale c'', d'^, etc., as compared with 
c'; i.e. continue the series of ratios given on p. 212 through another 

1 Pietro Blaserna, in his " Theory of Sound." 

2 Preyer places the lowest limit for some ears at 16 vibrations per second. 


7. What is the vibration number of a' in a scale in which c' (the key- 
note) = 256 vibrations ? 

8. The same singer may not be able to sing twice alike, i.e. in the 
same key ; how is it possible that the singing in both instances may be 
equally correct ? 

9. If one ear can hear a certain sound at 5 feet from the sounding 
body, and another ear at only 3 feet, how many times more sensitive is 
the former ear than the latter ? 

10. Why does the same bell always give a sound of nearly the same 
pitch ? 

11. a. What is the effect of striking a bell with different degrees of 
force ? b. What change in the vibrations is produced ? c. What property 
of sound remains the same ? 

12. a. Strike a key of a piano and hold it down ; what is the only 
change you observe in the sound produced, while it remains audible ? 
6. What is the cause of this change ? 

Section VII. 


178. Coexistence and siqjerjjosition of ivaves. — Interference. 
— When two or more currents of waves traverse the same 
medium at the same time and in the same direction, so that 
one set of waves is, as it were, superposed upon another, there 
are imparted to every particle of the medium simultaneously 
all the vibratory motions peculiar to the several waves. When 
two or more systems of waves act on a particle at the same 
time, they are said to interfere. The resultant motion of any 
particle at a given instant would be found on the principle of 
parallelogram of motions ; or, in case the several motions are 
parallel and occur in the same time, the resultant is the 
algebraic sum of the several motions. This will be best 
understood by means of graphical representations. In A 
(Fig. 173) are represented by dotted lines the wave lines of 
two coexisting currents of waves having the same wave-length 


and phase, but the amplitude of one greater than that of the 
other. For example, the amplitudes of the vibrations for the 
particle a are respectively a c 
and a e. Their algebraic sum 
is ad. In like manner the 
displacement of any particle 
of the medium traversed by 
the several wave currents at 
any instant is determined. 
The heavy line represents 
the form of the joint wave 
resulting from the combina- 
tion of the two. It will be 
seen that the only change is 
one of amplitude or inten- 

. In B are two wave-cur- 
rents whose waves are of 
the same length and ampli- 
tude, but with a difference 

of phase of 1 of a period, '^~-'' '"-^ ^^-''' ^^— '' 

. ^ . * ^ P Fig. 173. 

I.e. one is a quarter oi a 

wave-length behind the other. The result is a wave of the 

same length, but of different phase and amplitude. 

In C are two sets of waves of like length, of different 
amplitudes, and of opposite phases, or one is half a wave- 
length behind the other. The result is a set of waves of the 
same length, but diminished intensity. In D the conditions 
are the same as in C except that the components have the 
same amplitude. The result is that the two components de- 
stroy each other, and the particles of the medium are undis- 
turbed, as indicated by the straight line. 

In A (Fig. 174) are given two wave-currents whose wave- 
lengths are as 1 : -J- and whose phases in the beginning agree. 




The resultant of this combination with still another of ^ the 
wave-length of the longest is shown in B. In C is the same 

Fig. 174. 

combination as in A, but the phases differ by ^ of a period of 
the shorter wave. 

Fig. 175 represents wave-lines drawn by a vihrograph. The 
second line represents a sound two octaves above that which 


Fig. 175. 

the first line represents, and the third line shows the result 
that is produced by causing the fork to have two sets of 
simultaneous vibrations. 

Extend the arifi horizontally and cause the hand to move from 
side to side throuofh a wide arc and at the same time cause the 


hand to move in the same plane through shorter arcs in a sort of 
jerky movement. The result of these combined motions bears some 
faint resemblance to the motion of the tine as it draws line 3 above, 
or to the motion of an air particle as the tv^o currents of waves 
generated by the fork pass it. 

One may see a typical representation of superposed waves, or 
currents of small waves and wavelets creeping over the backs of 
larger ones and carving their surfaces into ragged and ever-changing 
outlines, in watching the billows of the sea, especially when the 
surface is swept by a breeze of varying intensity. 

In the diagrams given above only transverse vibrations are 
represented, bnt the results there depicted apply equally well 
to longitudinal vibrations and waves of condensations and 
rarefactions. In Fig. 176 the heavy line AB is a tyiykal 
representation of the resultant of two currents of aerial 

Fig. 176. 

sound-waves an octave apart, while the rectangular diagram 
C D is intended to represent a portion of a transverse section 
.of a body of air traversed by the joint wave corresponding to 
the heavy wave-line above. The depth of shading in different 
parts indicates the degree of condensation or rarefaction at 
those parts. 



Section YIII. 


179. Sonometer. — This instrument consists of two or more 
piano-wires of different thicknesses stretched lengthwise over 
a resonance box. One end of each wire is attached to the 
shorter arm of a bent lever, A or B (Fig. 177), and the tension 

Fig. 177. 

of the wire is regulated both by the lengths of the longer 
arms employed and by the magnitude of the weights suspended 
therefrom. The length of the vibrating portion of the strings 
is regulated by the sliding bridge C. 

Experiment 1. — Remove the bridge C, pluck one of the strings with 
the fingers at the middle point, causing it to vibrate as a whole, and note 
the pitch of the sound. Place the bridge under the same wire, and move 
it gradually toward one end of the sonometer, thereby shortening the 
vibrating portion ; the pitch rises as the vibrating portion is shortened. 
Vary the position of C until a pitch is obtained an octave above the pitch 
given at first when the entire wire was vibrating. It will be found that 
the length of the wire which gives the higher note is just half the original 
length; i.e. hy halving the wire its vibration-numher is doubled. At two- 
thirds its original length, it gives a note at an interval of a fifth above 
that given by its original length ; and generally the reciprocals of the 
fractions (p. 212) representing the relative vibration-numbers of the 
several notes of a scale represent the relative lengths of the wires that 
produce these notes. 

Now increase the tension of the wire ; the pitch rises. Increase the 
tension until the pitch has risen an octave; it will be found that the 
tension has been increased fourfold. 


Next try two wires whose lengths and tension are the same, but whose 
diameters are (say) as 1 : 2, and whose masses are consequently as 1 : 4 ; 
the pitch given by the wire of greater mass is an octave lower than the 
pitch given by the other wire. 

These conclusions may be summarized thus : The vibration- 
numbers of strings of the same material vary inversely as their 
lengths and the square roots of their masses per unit length, and 
directly as the square roots of their tensions. 

180. Stationary vibrations, nodes, etc. 

Experiment 2. — Hold one end of a rubber tube about 2 m long, while 
the other is fixed, and send along it a regular succession of equal pulses 
from the vibrating hand ; it will be easy, by varying the tension a little, 
to obtain a succes- ^ c 

sion of gauzy spindles ^ ^^,--'^*"™'llli^..^^^^^ 

(Fig. 178) separated 111^"^'^^. 

by points that are 

1 V ^ ^- Fig. 178. 

nearly or quite at rest. 

Unlike the earlier experiments, the waves here do not appear to travel 

along the tube ; yet in reality they do traverse it. The deception is 

caused by stationary points being produced by the interference of the 

advancing and retreating waves. 

This interference of direct and reflected waves gives rise 
to an important class of phenomena called stationary vibra- 
tions. The points of least motion, as a, b, and e (Fig. 178), are 
called nodes (from fancied resemblance to knots) ; the points 
of greatest amplitude, as d and c, are called antinodes ; and the 
portions between the nodes are called venters. 

In a similar manner a string may be made to vibrate in 3, 
4, etc., parts, as shown in C, D, and E (Fig. 179). The pitch 
of the tone produced by a string when it vibrates as a whole, 
as in A, is called the fundamental pitch of the string. The 
vibration frequency when the string divides into halves, as 
in B, is twice as great and consequently the pitch of the tone 
produced is an octave above that of the fundamental. Gen- 
erally the vibration frequency varies as the number of venters 
into which the string divides. 



Tones produced by a string or other body that vibrates in 
parts are called overtones or partial tones. If the overtones 

Fig. 179. 

harmonize (p. 223) with the fundamental of the vibrating 
body, they are called harmonics. 
181. Complex vibrations. 

Experiment 3. — Strike one of the lowest notes of a piano, hold the 
key down, and immediately apply the tip of the finger to some point of 
the wire struck, and notice any changes in tone that may occur after 
applying the finger. Kepeat this at many points along the string. If the 
fundamental sound disappear, there will probably be a sound of a higher 
pitch that will continue, showing that although you have stopped one set 
of vibrations, there were still other vibrations in the string of a higher 
vibration-period which you did not stop, and which now become audible 
since the louder fundamental is silenced. 

Experiment 4. — Press down the C'-key gently, so that it will not 
sound ; and while holding it down, strike the C-key strongly. In a few 
seconds release the key, so that its damper will stop the vibrations of the 
string that was struck, and you will hear a sound which you will recog- 
nize by its pitch as coming from the C'-wire. Place your finger lightly on 
the C'-wire, and you will find that it is indeed vibrating. Press down the 
right pedal with the foot, so as to lift the dampers from all the wires, 
strike the C-key, and touch with the finger the C'-wire; it vibrates. 
Touch the wires next to C, viz. B and T>' ; they have only a slight forced 
vibration. Touch G' ; it vibrates. 

ISTow it is evident that the vibrations of the C and G'-wires 
are sympathetic. But a C-wire vibrating as a whole cannot 
cause sympathetic vibrations in a C'-wire ; but, if it vibrates 


in halves, it may. Hence, we conclude that when the C-wire 
was struck it vibrated, not only as a whole, giving a sound of 
its own pitch, but also in halves ; and the result of this latter 
set of vibrations was, that an additional sound was produced 
by this wire, just an octave higher than the first-mentioned 

Again, the G'-wire makes 391.5 vibrations in a second, or 
three times as many (130.5) as are made by the C-wire ; hence 
the latter wire, in addition to its vibrations as a whole and in 
halves, must have vibrated in thirds, inasmuch as it caused 
the G'-wire to vibrate. It thus appears that a string may 
vibrate at the same time as a whole, in halves, thirds, etc., 
and the result is that a sensation is produced that is com- 
pounded of the sensations of several sounds of different pitch. 
A sound so simple that it cannot be resolved (see p. 227) is 
called a tone. 

182. Tones and notes. — A sound composed of many tones 
is called a note. 

Not onl}?- do stringed instruments produce notes, but no 
ordinary musical instrument is capable of producing a simple 
tone, i.e. a sound generated by vibrations of a single period. 
In other words, ivhen any note of any musical instrument is 
sounded, the7'e is produced, in addition to the primary tone, a 
number of other tones in a progressive series, each tone of the 
series being usually of less intensity than the preceding.^ The 
primary or lowest tone of a note is usually sufficiently intense 
to be the most prominent, and hence is called the fundamental 

Strings when struck produce many overtones, according to 
the place where they are struck, the nature of the stroke, and 
the density, rigidity, and elasticity of the string. 

1 The so-called Fourier theorem translated from the language of kinematics into 
that of acoustics asserts that " every regular musical sound is resolvable into a 
definite number of simple tones whose relative pitch follows the law of the partial- 
tone series." 



183. Beats. 

Experiment 5. — Strike simultaneously the lowest note of a piano and 
its sharp (black key next above), and listen to the resulting sound. 

You hear a peculiar wavy or throbbing sound, caused by an 
alternate rising and sinking in loudness. This phenomenon 
is still more conspicuous where the two lowest adjacent notes 
of a large organ are sounded together. Each recurrence of 
the maximum intensity is called Si beat. 

Let the continuous curved line A C (Fig. 180) represent a 
series of waves caused by striking the lower key, and the 

Fig. 180. 

dotted line a series of waves proceeding from the upper key. 
Now the waves from both keys may start together at A ; but 
as the waves from the lower key are given less frequently, so 
are they correspondingly longer ; and at certain intervals, as 
at B, condensations will correspond with rarefactions, pro- 
ducing by their interference momentary silence, too short, 
however, to be perceived ; but the sound as perceived by the 
ear is correctly represented in its varying loudness by the 
curved line A'B'C. 

It will be apparent from the study of Fig. 180 that exactly 
one beat will occur in each interval of time during which the 
acuter of two simple tones performs one more vibration than 
the graver tone. 

Hence the number of beats per second due to two simple tones 
is equal to the difference of their respective vibration numbers. 


The sensation produced on the ear by such a throbbing sound, 

when the beats are sufficiently frequent, is unpleasant, much 

as the sensation produced by flashes of light that enter the 

eye when you walk on the shady side of a picket fence is 

unpleasant. The unpleasant sensation called by musicians 

discord is due to beats. 

184. Origin of harmony and discord. — The harmonics in 

any note are produced successively by two, three, etc., times 

the number of vibrations made by its fundamental. Hence, 

if any two notes an octave apart, — for instance, C and C, — 

be sounded simultaneously, there will result for 

C, 1, 2, 3, 4, 5, 6, etc. ) ,. ^. x. , -x. ^- a 

^, „ , n , > times the number oi vibrations made 
C , 2, 4, 6, etc. ) 

by the fundamental of C ; so that the fundamental of C, and 

its overtones (with the exception of the highest overtones, 

which are too feeble to affect the general result), are in perfect 

unison with the overtones of C. Not only is there perfect 

agreement among the overtones of two notes an octave apart 

when sounded together, as when male and female voices unite 

in singing the same part of a melody, but the richness and 

vivacity of the sound are much increased thereby. 

Discord jDroduced by two sounds is explained by the fact 
that the sounds produce beats, which do not coalesce because 
the interval between them is too long. 

As the frequency of the beats increases, a point is finally 
reached where they cease to be recognized as distinct 
sounds and where they blend into a more or less pure 
tone. Beats may thus coalesce to produce beat-tones that 
are musical. 

It must not, however, be inferred that dissonance disap- 
pears immediately upon the intermittences becoming too rapid 
for individual recognition. If two tones form a narrower 
interval than a minor third, the combined sound is harsh and 
grating on the ear. 


Two tones must be in unison to produce absolutely perfect 
harmony. The nearest approach to it is an interval of an 
octave, and next in rank to the latter is a fifth. 

That two notes sounded together may harmonize, it is essential 
not only that the pitch of their fundamental tones he so widely 
different that they canjiot produce audible beats, but that no 
audible beats shall be formed by their overtones, or by an over- 
tone and a fundamental. 

Eor example, let, the vibration-numbers of the fundamentals 
of C and its octave C" be respectively 264 and 528 ; the num- 
ber of beats that they give is 264 in a second. If, instead of 
C", a note the vibration-number of whose fundamental is 527 
be sounded with C, the number of beats produced by their 
fundamentals would be 263, and no discord would result 
therefrom ; but there would be one beat per second between 
the first overtone of C and the fundamental of C", and this 
would introduce a discord. 

Observe that the relation between the vibration-numbers of 
the fundamentals of C and C, C and G, C and F, and C of 
any diatonic scale and any note in the same scale, can be 
expressed in terms of small numbers, e.g. 1:2, 2:3, 3:4, 
etc. (see p. 212). Generally, those notes and only those har- 
monize whose fundamental tones bear to one another ratios 
expressed by small numbers ; and the smaller the mmibers which 
exptress the ratios of the rates of vibration, the more -perfect is 
the harmony of two sounds. 

Not only may two notes whose relative vibration frequency 
is expressible by a simple ratio harmonize, but three or four 
may concur with the same result. A sound produced by the 
coexistence of three or more notes is called in music a chord. 
A consonant chord is a concord ; a dissonant chord is a discord. 

Fig. 181 is a graphical representation according to Helmholtz of 
the amount of dissonance contained in the several intervals of the 
diatonic scale. The intervals reckoned from C, are denoted by dis- 



tances measured along the horizontal straight line. The dissonance 
for each interval is represented by the vertical distance of the curved 
line from the corresponding point on the horizontal line. If we 
regard the outline as that of a mountain chain, the discords vv^ould 

"be represented by peaks, and the concords by passes, v^hile the 
steepness indicates the sharpness of deiinition of the interval. The 
calculations on which the curve is based are made from the notes 
produced by a violin. For piano-forte notes the curve would be 
slightly different. 

It follows, from what has been said, that only a limited 
number of notes can be sounded with any given note assumed 
as a prime without generating discord. Hence, the musical 
scale is limited to certain determinate degrees, represented 
by the eight notes of the so-called musical or diatonic scale. 
This scale is not the result of any arbitrary or fanciful 
arrangement, but is composed of notes selected because they 
harmonize with the prime of the scale, both as regards their 
fundamental tones and their overtones. 


1. Prepare a table of the series of overtones of C and G respec- 
tively, as on p. 223, and ascertain what overtones of the two series are in 

2. Arrange the notes in a single octave of the diatonic scale in the 
order of their rank with reference to their harmonizing with the prime 
of the scale, on the principle that "the smaller the numbers which ex- 


press the ratios of vibration, the more perfect is the harmony of two 

3. Verify your conclusions as follows : Strike the C-key of a piano, 
together with each of the seven white keys above it, consecutively, and 
compare the results of the different pairs with reference to harmony. 

Section IX. 


185. Simple sound-waves can differ only in length and 
amplitude; consequently tlie sounds which they produce can 
differ only in pitch and loudness. Complex sound-waves may 
differ, as we have seen, in /orm, and this gives rise to a prop- 
erty of sound called quality (by musicians, timbre). Quality 
is that property of sound, not due to pitch or intensity, that 
enables us to distinguish one sound from another. 

Although the variety of sounds one hears appears well-nigh 
infinite, yet no two sounds can differ from each other in any 
other respect than pitch, loudness, or quality. The length, 
amplitude, and form of the wave completely determine the 
wave, and these three elements of a wave are mutually inde- 
pendent, i.e. any one may be changed without altering the 
other two. Loudness depends on amplitude of vibrations, 
pitch on vibration-frequency, and quality on complexity of 
the motion of the vibrating particles. 

Let the same note be sounded with the same intensity, 
successively, on a variety of musical instruments, e.g. a violin, 
cornet, clarinet, accordion, jews-harp, etc. ; each instrument 
will send to your ear the same number of waves, and the 
waves from each will strike the ear with the same force, yet 
the ear is able to distinguish a decided difference between the 
sounds, — a difference that enables us instantly to identify 
the instruments from which they come. Sounds from instru- 
ments of the same kind, but by different makers, usually 


exhibit decided differences of character. For instance, of two 
pianos, the sound of one will be described as richer and fuller, 
or more ringing, or more ''wiry," etc., than the other. No 
two huinan voices sound exactly alike. 

Section X. 


186. Analysis of sound-waves. — The unaided ear is unable, 
except to a very limited extent, to distinguish the individual 
tones that compose a note. Helmholtz arranged a series of 
resonators of brass nearly 
spherical in shape, each hav- 
ing two openings; one, A 
(Fig. 182), large, for the re- 
ception of the sound-waves, 
and the other, B, small and ^Ijl 
funnel-shaped, and adapted 
for insertion into the ear. 
Each resonator of the series 
was adapted by its size to 
resound powerfully to only a 
single tone of a definite pitch. When any musical sound is 
produced in front of these resonators, the ear, placed at the 
orifice of any one, is able to single out, from the total number 
of tones composing the note, that overtone, if present, to 
which alone this resonator is capable of responding. By 
applying one resonator after another to the ear a sound is 
analyzed into its components. It is thus found, for instance, 
that the notes of a clarinet are composed only of the odd 
harmonics, or of tones whose vibration numbers are in the 
ratios of 1 : 3 : 5 : 7; and that the notes of a flute are substan- 
tially those of a tone and its octave. It is found that when a 


note is produced on a given instrument, not 6nly is there a 
great variety of intensity represented by the overtones, but 
all the possible overtones of the series are by no means 
present. Which are wanting depends very much, in stringed 
instruments, upon the point of the string struck. For example, 
if a string be struck at its middle point, no node can be formed 
at that point ; consequently, the two important overtones 
produced by 2 and 4 times the number of vibrations of the 
fundamental will be wanting. Strings of pianos, violins, etc., 
are generally struck near one of their ends, and thus they are 
deprived of only some of their higher and feebler overtones. 

" Every vowel is a particular quality of sound." The 
mouth cavity acts as a resonator and reflector. To each 
vowel corresponds a different form of resonating mouth cavity. 
Upon the number of upper overtones which are reinforced, 
and the relative intensities of the reinforcement, depends the 
quality of the vowel sound produced. Vowel sounds may be 
analyzed in a very interesting manner, as follows : Eaise the 
cover of a grand piano, press down the "loud pedal," and 
sing strongly some vowel sound, projecting the voice upon 
the exposed strings. When you cease to sing, that vowel will 
be repeated by the strings. Each component of the complex 
vibration will be taken up by that string in unison with it, 
and by noticing which strings vibrate a qualitative analysis of 
the sound is effected, 

187. Manomstric flames. — The pitch, intensity, and qual- 
ity of a sound may be studied at the same time by causing 
sound-waves to impinge directly upon some sensitive body 
without any intermediate process of selection. Apparatus 
like that shown in Eig. 183 will serve to illustrate. 

The cylindrical box A is divided by the membrane a into 
two compartments, c and h. Illuminating-gas is introduced 
into the compartment <?, through the rubber tube n, and burned 
at the orifice d. CD is a frame holding two mirrors, M, 



placed back to back, so that wMcliever side is turned toward 
the flame there is a reflection of the flame. 

When the mirror is at rest, an image of the flame will 
appear in the mirror as represented by A (Fig. 184). If the 
mirror be rotated, the flame appears drawn out in a band of 
light, as shown in B of the same figure. 

Fig. 183. 

Sing into the cone B (Fig. 183) the sound of oo in tool, and 
waves of air will run down the tube and beat against the 
membrane a, causing it to vibrate. The membrane has im- 
pressed upon it a complex motion resembling the original 
compound vibration of the vocal chords or other sounding 
body. The membrane in turn acts upon the gas in the com- 
partment G, throwing it into vibration. 



The result is, that instead of a flame appearing in the rotat- 
ing mirror as a continuous band of light, as B (Fig. 184), it is 
divided up into a series of tongues of light, as shown in C, 


Fig. 184. 

each condensation being represented by a tongue, and each 
rarefaction by a dark interval between the tongues. The 
number and size of the greater tongues indicate the fre- 
quency and amplitude of the fundamental vibration ; the 



subsidiary serrations correspond to the subsidiary vibrations 
or overtones. 

If a note an octave higher than the last be sung, we obtain, 
as we should expect, twice as many tongues in the same 
space, as shown in D. E represents the result when the two 
tones are produced simultaneously, and illustrates in a strik- 
ing manner the effect of interference. F represents the result 
when the vowel e is sung at the pitch of C ; and G, when 

Fig. 185. 

the vowel o is sung on the same key. These are called mano- 
metric flames. 

188. The phonautograph or phonograph. — Soiind-waves, 
however complex, may be caused permanently to record the 
succession and variation of their impulses, and thus, as it 
were, to inscribe their own autograph. Fig. 185 represents 
the original Edison phonograph. 

A metallic cylinder A is rotated by means of a crank. On 
the surface of the cylinder is cut a shalloAV helical groove 


running around the cylinder from end to end, like the thread 
of a screw. A small metallic point, or style, projecting from 
the under side of a thin metallic disk D (Fig. 186) which 

closes one orifice of the mouth- 
piece B, stands directly over the 
thread. By a simple device the 
cylinder, when the crank is turned, 
is made to advance just rapidly 
•^^^- ^^^- enough to allow the groove to 

keep constantly under the style. The cylinder is covered 
with tinfoil. The cone F is usually applied to the mouth- 
piece to concentrate the sound-waves upon the disk D. 

Kow, when a person directs his voice toward the mouth- 
piece, the aerial waves cause the disk D to participate in 
every motion made by the particles of air as they beat against 
it, and the motion of the disk is communicated by the style 
to the tinfoil, producing thereon impressions or indentations 
as it passes on the rotating cylinder. The result is that 
there is left upon the foil an exact representation of every 
movement made by the style. Some of the indentations are 
quite perceptible to the naked eye, while others are visible 
only with the aid of a microscope of high power. Fig. 187 
represents a piece of the foil as it would 
appear inverted after the indentations (here ^x^^^W^^;^^ 

greatly exaggerated) have been imprinted ^ 

upon it. 

The words addressed to the phonograph having been thus 
impressed .upon the foil, the mouth-piece and style are tem- 
porarily removed, while the cylinder is brought back to the 
position it had when the talking began, and then the mouth- 
piece is replaced. Now, evidently, if the crank be turned in 
the same direction as before, the style, resting upon the foil 
beneath, will be made to play up and down as it passes over 
ridges and sinks into depressions ; this will cause the disk D 


to reproduce the same vibratory movements that caused the 
ridges and depressions in the foiL The vibrations of the 
disk are communicated to the air, and through the air to the 
ear ; thus the words spoken to the apparatus may be, as it 
were, shaken out into the air again at any subsequent time, 
even centuries after, accompanied by the exact accents, into- 
nations, and quality of sound of the original 

Subsequently Edison improved this instrument by replacing the 
metallic foil by a cylinder of hard wax composition, rotating it by 
an electric motor, and providing an improved form of style which 
engraves upon the wax the most delicate variations of vibratory 
motions, and thus, as it were, reproduces speech and musical notes 
with all their delicate shades of expression and modulation. In its 
improved form it has become a commercial instrument, and is used 
in some cases in the place of stenography, the correspondence being 
dictated to the instrument and then rej)roduced by means of a 

189. Synthesis of sound-waves. — The sound of a tuning- 
fork when its fundamental is reenforced by a suitable reso- 
nance-cavity, is very nearly a simple tone. 

If two nioiuited forks forming the interval of an octave be 
sounded together, the tones proceeding from the separate 
forks soon blend together into one sound, to which we assign 
the pitch of the lower fork, and a quality richer than that of 
either. So strong is the illusion, that we cannot believe the 
higher fork to be sounding, until we ascertain that placing a 
finger on its prongs so as to damp its vibrations at once 
changes the timbre, reducing it to the dull, uninteresting 
quality of a simple tone. If to these two forks there be 
added a fork whose interval is a fifth above the higher of 
the first two (i.e. one which gives the second harmonic of the 
first), the three tones blend as perfectly as did those of the two 
forks ; the only difference perceptible being an additional 
increase of richness. 


By sounding simultaneously several forks of different but 
appropriate pitch, and with the requisite relative intensities, 
Helmholtz succeeded in producing sounds peculiar to various 
musical instruments, and even in imitating most of the vowel 
sounds of the human voice. 

Thus it appears that he has been able to determine, both 
analytically and synthetically, that the quality of a given sound 
depends upon what overtones combine with its fundamental tone, 
and upon their relative intensities ; or, more briefly, upon the 
form of the sound-wave, since the form must be determined 
by the character of its components. 

Section XI. 


190. Classification of musical instruments. — Musical in- 
struments may be grouped into three classes : (1) stringed 
instruments ; (2) wind instruments, in which the sound is 
due to the vibration of columns of air confined in tubes ; (3) 
instruments in which the vibrator is a membrane or plate. 
The first class has received its share of attention ; the other 
two merit a little further consideration. 

191. Wind instrume7its. 

Experiment 1. — Fig. 188 represents a set of Quincke's whistles. The 
tubes are of the same size, but of varying length. Blow through the 
small tube across the lips of the large tube of each whistle in the order of 
their lengths, commencing with the longest. 

Repeat the experiment, closing the end of the whistle farthest from 
you with a finger, so as to make what is called a " closed pipe." i 

The pitch of vibrating air-columns, as well as of strings, 

1 The eif ect of interference is well shown with these tubes by blowing at the same 
time through two tubes of nearly the same length. A peculiar rattling sound is the 
result. A very different result is obtained when two tubes a fif'th anart are sounded 


varies with the length, and (1) in both stopped'^ and open'^ 
pipes the number of vibrations is inversely proportional to the 
length of the pipe. (2) An open pipe gives a note an octave 
higher than a closed pipe of the same length. 

Fig. 188. 

Experiment 2. — Take some of the longer whistles, and blow as before, 
gradually increasing the force of the current. It will be found that only 
the gentle current will give the full musical fundamental tone of the tube, 
— a little stronger current producing a mere rustling sound ; but when 
the force of the current reaches a certain limit, an overtone will break 
forth ; and, on increasing still further the power of the current, a still 
higher overtone may be reached. 

1 The terms " stopped " and " open " apply to only one end of the pipe ; the other, 
in both kinds, is always open. 



Fig. 189 represents an open organ-pipe provided with a 
glass window A in one of its sides. A wire hoop B has 
stretched over it a membrane, and the whole is suspended by 
a thread within the pipe. If the membrane be 
placed near the upper end, a buzzing sound pro- 
ceeds from the membrane when the fundamental 
tone of the pipe is sounded; and sand placed on 
the membrane will dance up and down in a lively 
manner. On lowering the membrane, the buzzing 
sound becomes fainter, till, at the middle of the 
tube, it ceases entirely, and the sand becomes 
quiet. Lowering the membrane still further, the 
sound and dancing recommence, and increase as 
the lower end is approached. 

(3) When the fundamental tone of an open pipe 
is p)roduced, its air-column divides into two equal 
vibrating sections, with the anti-nodes at the extrem- 
ities of the tube, and a node in the middle. 

If the pipe be stopped, there is a node at the 
stopped end ; if it be open, there is an anti-node 
at the open end; and in both cases there is an 
anti-node at the end where the wind enters, which 
is always to a certain extent open. 
A, B, and C of Fig. 190 show respectively the positions of 
the nodes and anti-nodes for the fundamental tone and first 
and second overtones of a closed pipe ; and A', B', and C 
show the positions of the same in an open pipe of the same 
length. The distance between the dotted lines shows the 
relative amplitudes of the vibrations of the air-particles at 
various points along the tube. Now the distance between a 
node and the nearest anti-node is a quarter of a wave-length. 
Comparing, then, A and A', it will be seen that the wave- 
length of the fundamental of the closed pipe must be tvv^ice 
the wave-length of the fundamental of the open pipe ; hence 

Fig. 189. 



the vibration period of the latter is half that of the former; 
consequently the fundamental of the open pipe must be an 
octave higher than that of the closed pipe. 

In the three cases (A, B, and C) of the closed tube, the 
length of the air-column is divided into i, |, and | segments 
respectively ; hence the corresponding vibration-numbers are 




1 / 


\ 1 



\ / 
\ / 


1 / 
\ / 

1 \ 
1 \ 

\ 1 
\ / 

/ \ 

/ \ 

1 \ 
1 \ 

\ 1 

/ , 
' 1 

Fig. 190. 

as 1 : 3 : 5, etc. Hence, (4) m a closed tube, only those over- 
tones tvhose vibration-numbers correspond to the odd multiples 
of that of the fundamental are present. 

In the cases of the open tube, the length of the air-column 
is divided into |, |, and | segments respectively ; their vibra- 
tion-numbers are therefore as 1 : 2 : 3, etc. Hence, (5) in an 
open tube, the complete series of overtones corresponding to its 
fundamental may be present. 

192. Sounding plates, etc. 

Experiment ^.—Fasten with a screw the elastic brass plate A (Fig. 
191) on the upright support. Strew writing-sand over the plate, draw a 



rosined bass bow steadily and firmly over one of its edges near a corner, 
and at the same time touch the middle of one of its edges with the tip 
of the finger ; a musical sound will be produced, and the sand will dance 
up and down, and quickly collect (1) in two rows, extending across the 
plate at right angles to each other. Draw the bow across the middle of 

Fig. 191. 

an edge, and touch with a finger one of its corners ; the sand will arrange 
itself in two diagonal rows (2) across the plate, and the pitch of the note 
will be a fifth higher. Touch, with the nails of the thumb and fore- 
finger, two points a and 6 (3) on one edge, and draw the bow across the 
middle c of the opposite edge, and you will obtain additional rows and a 
shriller note. 

By varying the position of the point touched and bowed, 
a great variety of patterns can be obtained, some of which 
are represented in Fig. 192. It will be seen that the effect 
of touching the plate with a finger is to prevent vibration at 
that point, and consequently a node is there produced. The 
whole plate then divides itself up into segments with nodal 
division lines in conformity with the node just formed. The 
sand rolls away from those parts which are alternately thrown 
into crests and troughs, to the parts that are at rest. 



193. Interference. 

Experiment 4. — C (Fig. 191) is a tin tube made in two parts that 
telescope one within the other. The extremity of one of the parts ter- 
minates in two slightly smaller branches. Bow the plate, as in experi- 
ment 3 (1), place the two orifices of the branches over the segments 
marked with the + signs, and regulate the length of the tube so as to 
reenforce the note given by the plate, and set the plate in vibration. 



1 J r 

J i L 

Fig. 192. 

Now turn the tube around, so that one orifice may be over a + segment, 
and the other over a — segment ; the sound due to resonance entirely 
ceases. It thus appears that the two segments marked + pass through 
the same phases together ; likewise the phases of — segments correspond 
with one another ; i. e. when one + segment is bent upward, the other 
is bent upward, and at the same time the two — segments are bent 
dovraward ; for, when the two orifices of the tube are placed over two 
+ segments or two — segments, two condensations followed by two rare- 
factions pass up these branches and unite at their junction to produce a 
loud sound ; but when one of the orifices is over a + segment, and the 



Fig. 193. 

other over a — segment, a condensation passes up one branch at the 
same time that a rarefaction passes up the other, and the two destroy 
each other when they come together; i.e. the two sound-waves combine 
to produce silence. 

194. Bells. — A bell or goblet is Subject to the same laws 
of vibration as a plate. 

Experiment 5. — Nearly fill a large goblet with 
water, strew upon the surface lycopodium powder, 
and draw a rosined bow gently across the edge of 
the glass. The surface of the water will become 
rippled with wavelets (Fig. 193) radiating from four 
points 90° apart, corresponding to the centers of 
four venters into which the goblet is divided, and 
the powder will collect in lines proceeding from the 
nodal points of the bell. By touching the proper 
points of a bell or glass with a finger-nail, it may be 
made to divide itself, like a plate, into 6, 8, 10, etc. 

(always an even number), vibrating parts. 

Experiment 6. — Remove the brass plate (Fig. 191) from its support, 

and fasten the bell B (Fig. 194) on the support. Bow the edge of the 

bell at some point, and hold the open 

tube C in a horizontal position with the 

center of one of its openings near that 

point of the edge of the bell which is 

opposite the point bowed. The tube 

loudly reenforces the sound of the bell. 

Move the tube around the edge of the 

bell and find its nodes. 

Thrust the plunger I) into the open 

end E of the tube, and find what part of 

the length of an open tube a closed tube should be to reenforce a sound 

of a given pitch. 

195. Vocal organs. — It is difficult to say wliich is more to 
be admired, the wonderful capability of the human voice or 
the extreme simplicity of the means by which it is produced. 
The organ of the voice is a reed instrument situated at the 
top of the windpipe, or trachea. A pair of elastic bands, a a 
(Fig. 195), called the vocal chords, is stretched across the 


Fig. 194. 


top of the windpipe. The air-passage b between these chords 
is freely open while a person is breathing ; but when he 
speaks or sings, they are brought together so as to form a nar- 
row, slit-like opening, thus making a 
sort of double reed, which vibrates 
when air is forced from the lungs 
through the narrow passage, some- 
what like the little tongue of a toy 
trumpet. The sounds are grave or 
high according to the tension of the 
chords, which is regulated by muscu- 
lar action. The cavities of the mouth 
and the nasal passages form a com- 
pound resonance - tube. This tube 

■^ _ JblG. 195. 

adapts itself, by its varying width and 

length, to the pitch of the note produced by the vocal chords. 
Place a finger on the protuberance of the throat called 
"Adam's apple," and sing a low note; then sing a high note, 
and you will observe that the protuberance rises in the latter 
case, thus shortening the distance between the vocal chords 
and the lips. Set a tuning-fork in vibration, open the mouth 
as if about to sing the corresponding note, place the fork in 
front of it, and the cavity of the mouth will resound to the 
note of the fork, but will cease to do so when the mouth 
adapts itself to the production of some other note. The 
different qualities of the different vowel sounds are produced 
by the varying forms of the resonating mouth-cavity, the 
pitch of the fundamental tones given by the vocal chords 
remaining the same. This constitutes articulation. 

Strictly speaking, consonants are not distinct sounds or the rep- 

- resentatives of sound. They represent rather, when they precede 

vowels, the different positions of the organs of speech from which 

(like spring-boards, as it were) the vowels are attacked. i Following 

1 A commoii fault with young singers is attempting to " sing the consonants." 



vowels, they represent the position of the organs of speech at the 
interruption of the vowel sounds, and the consequent modifications 
of these sounds. Consonants are accordingly classified into labials, 
dentals, gutturals, etc. The more care exercised in placing the 
organs in suitable positions for attack or interruption and the less 
sound emitted from these points at the moment of attack, the 
clearer is the articulation. Modulations of the voice in conversa- 
tion take place usually in musical intervals. Singing differs from 
speaking chiefly in the manner in which the vocal sounds are modi- 
fied. In both, the sustained sounds are vowel sounds. In fact 
only vowel sounds are musical, and any language is musical in 
proportion to the number of vowels it contains. Thus Greek is a 
more musical language than Latin, and Italian than German. 

196. The ear. — In Fig. 196, A represents the external ear- 

FlG. 196. 

passage; (x is a membrane, called the tympanum, stretched 
across the bottom of the passage, and thus closing the orifice 
of a cavity b, called the drum; c is a chain of small bones 
stretching across the drum, and connecting the tympanum 


with the thin membranous wall of the vestibule e ; ff are a 
series of semicircular canals opening into the vestibule ; g is 
an opening into another canal in the form of a snail-shell g\ 
hence called the cochlea (this is drawn on a reduced scale) ; 
c? is a tube (the Eustachian tube) connecting the drum with 
the throat; and h is the auditory nerve. The vestibule and 
all the canals opening into it are filled with a transparent 
liquid. The drum of the ear contains air, and the Eustachian 
tube forms a means of ingress and egress for air through the 

Now how does the ear hear ? and how is it able to dis- 
tinguish between the infinite variety of form, rapidity, and 
intensity of aerial sound-waves so as to interpret correctly 
the corresponding quality, pitch, and loudness of sound ? 
Sound-waves enter the external ear-passage A as ocean-waves 
enter the bays of the sea-coast, are reflected inward, and 
strike the tympanum. The air-particles, beating against this 
drum-head, impress upon it the precise wave-form that is 
transmitted to it through the air from the sounding body. 
The motion received by the drum-head is transmitted by the 
chain of bones to the membranous wall of the vestibule. 
From the walls of the spiral passage of the cochlea project 
into its liquid contents thousands of fine elastic threads or 
fibres, called "rods of Corti." As the passage becomes smaller 
and smaller, these vibratile rods become of gradually dimin- 
ishing length and size (such as the wires of a piano may 
roughly represent), and are therefore suited to respond sym- 
pathetically to a great variety of vibration-periods. This 
arrangement is sometimes likened to a " harp of three thou- 
sand strings " (this being about the number of rods). The 
auditory nerve at this extremity is divided into a large num- 
ber of filaments, like a cord unraveled at its end, and one 
of these filaments is attached to each rod. Now, as the 
sound-waves reach the membranous wall of the vestibule, 


they set it, and by means of it the liquid contents, into fo7xed 
vibration, and so through the liquid all the fibers receive an 
impulse. Those rods whose vibration periods correspond 
with the periods of the constituents forming the compound 
wave are thrown into sympathetic vibration. The rods stir 
the nerve filaments, and the nerve transmits to the brain the 
impressions received. Much as a piano when its dampers are 
raised and a person sings into it, may be said to analyze each 
sound-wave, and show by the vibrating strings of how many 
tones it is composed, as well as their respective pitches, and by 
the amplitude of their vibrations their respective intensities ; 
so, it is thought, this wonderful harp of the ear analyzes 
every complex sound-wave into a series of simple waves. 
Tidings of the disturbances are communicated to the brain, 
and there, in some mysterious manner, these disturbances are 
interpreted as sound of definite quality, pitch, and intensity. 

PART 11. 


Section I. 


In the preceding pages the theory of heat has been several 
times anticipated ; we are now better qualified to judge of its 

197. Energy of mass motion convertible into heat. 

Experiment 1. — Hold some small steel tool upon a rapidly revolving 
dry grindstone; a shower of sparks flies from the stone. Place a ten- 
penny nail upon a stone and hammer it briskly ; it soon becomes too hot 
to be handled with comfort, and we may conceive that if the blows were 
rapid and heavy enough, it might soon become red hot. Eub a desk 
with your fist, and your coat-sleeve with a metallic button ; both the 
rubbers and the things rubbed become heated. 

Yon observe that in every case heat is generated at the 
expense of work or mass energy ; i.e. mass energy destroyed 
becomes heat. When the brakes are applied to the wheels of 
a rapidly moving railroad train, its energy is converted into 
heat, much of which may be found in the wheels, brake- 
blocks, and rails. The meteorites, or " shooting-stars," which 
are seen at night passing through the upper air, sometimes 
strike the earth, and are found to be stones heated to a light- 
giving state. They become heated when they reach our 
atmosphere, in consequence of their motion being checked by 
the resistance of the air. 



198. Heat convertible into mass energy. 

Experiment 2. — Take a tliin glass flask A (Fig. 197), half fill it witli 
water, and fit a cork air-tight into its neck. Perforate the cork, insert a 
glass tube . bent as indicated in the figure, and extend it into the water. 
Apply heat to the flask; soon the liquid rises in the 
Q u u tube, and flows from its upper end. 

Here heat produces mechanical motion, and 
does work in raising a mass in opposition to 
gravitation. Every steam engine is a heat en- 
gine, i.e. the power of steam is due to its heat. 
The steam which leaves the cylinder of an en- 
gine, after it has set the piston in motion, is 
cooler than when it entered. 

It will be shown hereafter that in all cases 
when work is done by heat without waste or 
loss, the quantity of heat consumed is propor- 
tional to the work done ; and, conversely, by 
the performance of a definite quantity of work 
an equivalent quantity of heat is produced; in 
other words, there is a definite quantitative 
relation between heat and work. 

If heat be consumed, and mechanical work thereby per- 
formed, we are justified in saying that heat has transformed 
itself into mass energy ^ and, conversely, if mass energy be 
expended and heat thereby produced, we may say that mass 
energy has transformed itself into heat. 

Now, when the appearance of one thing is so connected 
with the disappearance of another that the quantity of the 
thing produced can be calculated from the quantity of that 
which disappears, we conclude that the one is formed at the 
expense of the other, and that they are only different forms 
of the same thing. We have, therefore, reaison to believe 
that heat is of the same nature as mass energy ; i.e. it is only 
another form of energy. 

Fig. 197. 


199. Theory of heat. — A body loses motion in communi- 
cating it. The hammer descends and strikes the anvil ; its 
motion ceases, but the anvil is not sensibly moved ; the only 
observable effect produced is heat. Instead of a motion of 
the hammer and anvil, there is now an increased vibratory 
motion of the molecules that compose the hammer and anvil, 
— simply a change from molar to molecular motion. Of course, 
this latter motion is invisible. According to this view, heat 
is hut a name for the energy of vibration of the Tnolecules of a 
body, or, briefly, heat is molecular kinetic energy. The 
science which treats of heat as a form of energy is called 

A body is heated by having the motion of its molecules 
quickened, and cooled by parting with some of its molecular 
motion. Cold is comparable to rest, heat^ to motion. One 
body is hotter than another when the average kinetic energy of 
the molecides in it is greater than in the other. 

As late as the beginning of tlie present century heat was gen- 
erally regarded as " a sensation which the presence of fire" (an 
" igneous ^uicZ," "matter of heat," called sometimes " caloric ") 
" occasions in animate and inanimate bodies." A text-book of that 
period makes this significant statement : " There is fire in the wood, 
and there is air in the field, though we do not perceive either while 
at rest. Eubbing two pieces of wood does not create fire any more 
than the blowing of the wind creates air. Motion renders both 
perceptible.'''' The former and the more modern views are in har- 
mony in attributing the immediate cause of the sensation to motion. 
According to the former view, the sensation is produced by putting 
an imaginary fluid in motion; according to the modern view it is 
produced by quickening the motion of the molecules of a body. 

The material theory became untenable when it was shown by 
Count Eumford ^ that the quantity of heat that may be evolved by 

1 Whenever there is occasion to speak of the sensation which heat is capable of 
producing, it should never be called heat, but it should be termed a heat sensation or 
the sensation of heat. 

2 The great discovery of the non-materiality of heat was made by an American, 
Benjamin Thompson (Count Eumford), then (1798) residing in Munich. This dis- 


friction, as, for instance, in the boring of cannon, is practically 
limitless, or is limited only by the mechanical power available. 
Now according to this theory, when a piece of metal is rubbed the 
caloric is rubbed or squeezed out of it ; but, as Rumford argued, 
" anything which a body can continue to furnish without limitation 
cannot possibly be a material substance." At about the same time 
Davy showed that two pieces of ice may be melted by rubbing them 
together in a space whose temperature is below the melting point. 

200. Heat, the lowest for m of energy. — Heat is often spoken 
of as the '' lowest form of energy." That is, all other forms 
of kinetic energy tend to transform themselves into the 
"lower" form of heat ; as water tends to seek a lower level. 
When energy is spent in doing work, that portion which 
appears in no other form appears as heat. 

Section II. 


201. Mechanical eiiergy a source of heat. — As heat is 
energy, so all heat must originate in some form of energy, 
i.e. by the transformation of some other form of energy into 

In the preceding section it was shown that heat may be 
generated at the expense of molar motion, i.e. molar motion 
checked usually results in molecular motion, the energy of 
which is heat. By friction, by compression, by percussion, 
or by any process by which mass motion is arrested, heat is 
mechanically generated. 

202. Chemical union a source of heat. 

Experiment 1. — Take a glass test-tube half full of cold water and pour 
into it one-fourth its volume of strong sulphuric acid. The liquid almost 
instantly becomes so hot that the tube cannot be held in the hand. 

covery lies at tlie foundation of the dynamical theory of heat, and directly led to the 
grandest doctrines of modern science, the correlation and the conservation of energy. 
(See p.304 ) 


When water is poured upon quicklime, heat is rapidly 
developed. The invisible oxygen of the air combines with 
the constituents of the various fuels, such as wood, coal, oils, 
and illuminating-gas, and gives rise to what we call burning, 
or co77ihustion, by which a large amount of heat is generated. 
In all such cases the heat is generated by the combination or 
clashing together of molecules of substances that have an 
affinity (i.e. an attraction) for one another. Before union 
the energy of the molecules is of the same kind as that of a 
stone on a shelf. When the shelf is withdrawn, gravity con- 
verts the potential energy of the stone into kinetic energy ; 
so affinity converts the potential energy of the molecules into 
kinetic energy of vibration, i.e. into heat. 

When a definite mass, say of carbon or hydrogen, is burned, 
the quantity of heat produced is definite ; hence the different 
fuels have a definite heat value, which depends in part upon 
the combustion equivalents of their constituents. 

In a majority of cases chemical union is attended by the evolu- 
tion of heat ; but in some cases work has to be done by heat upon 
separate elements to force them to combine either directly or indi- 
rectly ; hence in such cases the union is attended by a consumption 
or disappearance of heat, and the decomposition of compounds thus 
formed is attended by an evolution of heat. 

203. Origin of animal heat and muscular motion. — The 
plant finds its food in the air (principally the carbonic acid 
in the air) and in the earth, in a condition analogous to that 
of a fallen weight ; but, by the agency of the sun's radiation, 
work is performed upon this matter during the growth of the 
plant ; potential energy is stored in the plant, — the weight 
is drawn up as it were. The animal now finds its food in the 
plant, appropriates the potential energy stored in the plant, 
and, by chemical action, chiefly by the union of carbon and 
hydrogen with oxygen, this energy is converted into the en- 
ergy of motion in the form of heat and muscular motion, — 


the weight falls and its energy becomes kinetic. The plant, 
then, may be regarded as a machine for converting energy 
of motion received from the sun into potential energy ; the 
animal, as a machine for transforming it again into energy 
of motion. 

204. The sun as a source of energy. — The sun is not only 
the source of the energy exhibited in the growth of plants, as 
well as of the muscular and heat energy of the animal, but is 
also the source, directly or indirectly, of very nearly all the 
energy employed by man in doing work. Our coal-beds, the 
results of the deposit of vegetable matter, are vast store- 
houses of the sun's energy, rendered potential during the 
growth of the plants many ages ago. Every drop of water 
that falls to the earth and rolls its way to the sea, contribut- 
ing its mite to the immense water-power of the earth, and 
every wind that blows, derives its power directly from the 

205. Dissipatio7i of energy. 

Work is done by heat only when it passes from a higher to a 
lower level, i.e. from a higher temperature to a lower temperature. 
In other words heat would not be available for doing work if all 
matter were reduced to the same temperature. Heat has a ten- 
dency to become uniformly diffused. The temperature of the entire 
material universe tends to uniformity. This does not imply that 
the quantity of energy in the universe changes, but it does imply 
that the quantity of energy available to man for doing work is 
diminishing. A railroad car with its furniture and all objects on 
board the car may be considered as constituting a system of bodies. 
If the car be in rapid motion, it possesses a large quantity of energy 
and each body in the system possesses energy proportionate to its 
mass, but the energy of no one of these bodies is available for doing 
work upon another. Why ? Because all have like velocities, and 
no one can impart velocity to another. So in a system where all 
the molecules have a like velocity, there can be no transfer of 
velocity or energy, and hence no work can be done within the 
system by any of its members. 


Now since all forms of energy tend toward this lower form of 
energy, and the temperature of all matter tends towards a uni- 
formity, and since in proportion as uniformity of temperature is 
reached the quantity of available energy in the universe is dimin- 
ished, we are forced to the conclusion expressed by Tait that " The 
available energy of the universe tends to zero." ^ This. is called by 
Lord Kelvin the Doctrine of Dissipation of Energy. 

Section III. 


206. Temjperatuve defined. — The words ivarm, hot, cool, 
cold are associated in our minds with a series of sensations 
which we suppose to indicate a corresponding series of states 
of an object with respect to heat; that is, the agent which 
produces these sensations is heat. These are all temperature 
terms, and refer to the state of an object with reference to 
heat. Tertijiievature is the state of matter in respect to heat. 
When the quantity of heat in a body increases, its tempera- 
ture is said to rise ; and when this diminishes, its temperature 
is said to fall. The relation which temperature bears to heat 
is analogous to that which hydrostatic pressure bears to 
water. Water flows from high level to low level ; heat flows 
from high temperature to low temperature. When we pour 
water into a vessel, the level rises ; so heat entering a body 
raises its temperature (unless it is transformed in doing work). 
It takes more water to fill a large vessel to a given depth 
than a small one ; it takes more heat to raise the temperature 
of a body of large mass a certain amount than to raise the 
temperature of a smaller mass of the same substance an equal 

1 "In the beginning " poiaits to a far distant period when all the energy of the 
physical universe was in the available form. Physical science foresees a time when 
all available energy shall become zero and all the processes of nature must cease. 
" The marvellous mechanism of nature will then have run down, and no further 
motion or life-process will be possible unless some new order intervenes of which 
we have no knowledge or conception." 


amount. Depth of water in vessels of varying size is not 
proportional to the quantity of water they contain ; tempera- 
ture of bodies of varying mass is not necessarily proportional 
to the quantity of heat they contain. For example, a pint of 
water at a given temperature does not contain the same quan- 
tity of heat as a gallon of water at the same temperature. 

If body A be brought in contact with body B, and A tend 
to impart heat to B, then A is said to have a higher tempera- 
ture than B. Temperature is sometimes spoken of as the state 
of a body ivith reference to its tendency to commuiiicate heat to, 
or receive heat from, other bodies. The direction of the flow 
of heat determines which of two bodies has the higher tem- 
perature. If the temperature of neither body rise at the 
expense of the other, then both have the same temperature, 
or are said to be in thermal eqidlibrium. 

Temperature depends on the average kinetic energy of the 
molecules. The temperature of a substance increases propor- 
tionally to the mean square of the velocity of vibration of its 
molecules. Bodies have the same temperature when the 
average energy of the molecules of each is the same. 

207. Temperature a relative term. 

As the term is now used, temperature is a relative expression, 
not an expression of absolute quantity. It is regarded rather as a 
quality capable of greater or less intensity, than as a quantity which 
may be added to or subtracted from other quantities of the same 

For instance, if the temperatures of two bodies be respectively 
25° C. and 35° C, we say with truth that their temperatures differ 
by 10 centigrade degrees ; but we cannot say with proprietj^ that a 
temperature of 25° subtracted from a temperature of 35° leaves a 
temperature of 10°, or that the two temperatures added together 
give a temperature of 60°. ^ 

There is no propriety in the expression "twice as hot." The 
term temperature is similar to the term hardness, insomuch as we 
are able to construct a scale for both, so that a body may have a 
definite place in the scale and be less hot or less hard than some- 


thing above it, and yet we are not able to estimate either tempera- 
ture or hardness quantitively. 

Section IV. 


208. Use of a thermometer. — A thermometer is an instru- 
ment for indicating temperature, i.e. tlie difference between 
the temperature of a given body and some standard tempera- 
ture (§ 211). For the difference between two standard tem- 
peratures, such as the melting point of ice and the boiling 
point of water, is one capable of accurate subdivision into 
any number of equal parts, which form successive equal steps 
from the lower to the higher temperature. 

209. TeTuperature indicated by expansion. — The effects of 
expansion by heat are well illustrated in the common ther- 
mometer. As its temperature rises, both the glass and the 
mercury expand ; but, as liquids in general are more expansi- 
ble than solids, the mercury expands much more rapidly than 
the glass, and the apparent expansion of the mercury, shoivn hy 
its rise in the tube, is the difference between the actual increase of 
volume of the mercury and that of the capacity of the glass ves- 
sel containing it. The thermometer, then, primarily indicates 
changes of volume ; but as changes of volume in this case are 
caused by changes of temperature, it is commonly used for 
the more important purpose of indicating temperature. 

If a thermometer be brought into intimate contact with a 
body whose temperature is sought, as, for instance, a liquid 
into which it is plunged, or the air in a room, the mercury in 
the tube rises or falls until it reaches a certain point, at 
which it remains stationary. We then know that it is in 
thermal equilibrium with the surrounding body. Hence the 
reading, as it is called, of the thermometer indicates not only 
the temperature of the mercury, but of the surrounding body. 


210. Construction of a thermometer. — A thermometer con- 
sists generally of a glass tube of capillary ^ bore, terminating 
at one end in a bulb. The bulb and part of the tube are filled 
with, mercury, and the space in the tube above the mercury 
is a partial vacuum. On the tube, or on a plate of metal 
behind the tube, is a scale to show the hight of the mercurial 

211. Standard temjper attires. — That a thermometer may 
indicate any definite temperature, it is necessary that its 
scale should relate to some definite and unchangeable points 
of temperature. Fortunately Nature furnishes us with two 
convenient standards. It is found that under ordinary atmos- 
pheric pressure ice always melts at the same temperature, 
called the melting iDoint. Again, the temperature of steam 
rising from boiling water under the same pressure is always 
the same. 

212. Graduation of thermometers. — First, the bulb of a 
thermometer is placed in melting ice (Fig. 198) and allowed 
to stand until the surface of fhe mercury becomes stationary, 
and a mark is made upon the stem at that point, which indi- 
cates the melting point. Then the instrument is suspended 
in steam rising from boiling water (Fig. 199), so that all but 
the very top of the column is in the steam. 

The bulb is placed in a metallic vessel, M, with a narrower 
upper part, A. This narrower part is surrounded by a larger 
part, B. By observing the arrows it is seen that steam sur- 
rounds the inner part, and thus prevents its cooling ; it 
escapes by the tube D. The orifice of D is large enough to 
allow the steam to escape freely, and thus prevent a pressure 
inside the vessel greater than the atmospheric pressure. To 
guard against such a contingency a pressure gauge m is in- 
serted ill the vessel. The liquid in both arms of the gauge 

1 In consequence of refraction of light at the cylindrical surface of the glass tube, 
the diameter of the cylinder of mercury in the bore appears magnified. 



must be kept at the same level througliout the operation. 
The mercury rises in the stem of the thermometer until its 
temperature becomes the same as that of the steam, when it 
becomes stationary. A barometer is consulted, and due allow- 
ance having been made for atmospheric pressure at the time, 
a mark is placed on the stem to indicate the boiling point. 
This boiling point is the temperature of steam at a pressure 

Fig. 198. 

Fig. 199. 

of 760 mm of mercury at 0° C. in the latitude of Paris (48° 
50'), 60 meters above sea-level. Then the space between the 
two points found is divided into a convenient number of 
equal parts (provided that the bore of the tube is of uniform 
diameter) called degrees, and the scale is extended above and 
below these points as far as is desirable. 

Two methods of division are adopted in this country (see a 
and h, Fig. 200) : by one, the space is divided into 180 equal 
parts, and the result is called the Fahrenheit scale, from the 
name of its designer ; by the other, the space is divided into 







100 equal parts, and the resulting scale is called centigrade, 
-which, means one hundred steps. In the Fahrenheit scale, 
which is generally employed for ordinary household purposes, 
^ the melting and boiling points are marked 

a respectively 32° and 212°. The of this 

scale (32° below melting point), which is 
about the lowest temperature that can be ob- 
tained by a mixture of snow and salt, was 
incorrectly supposed to be the lowest temper- 
ature attainable. The centigrade scale, which 
is generally employed by scientists, has its 
melting and boiling points more conveniently 
marked, respectively 0° and 100°. A temper- 
ature below 0° in either scale is indicated 
by a minus sign before the number. Thus, 
- 12° F. indicates 12° below 0° (or 44° below 
melting point), according to the Fahrenheit 
scale. Under F. and C. (in the left column. 
Fig. 201), the two scales are placed side by 
side, so as to exhibit at intervals a compara- 
tive view. The Fahrenheit and centigrade 
scales agree at — 40°, but diverge both ways 
from this point. 

213. Conversion from one scale to the other. 
— Since 100° C. =c= 180° F., 5° C. =^ 9° F., 
or 1° C. =crr I of 1° F. Hence, to convert 
Fig 200 centigrade degrees into Fahrenheit degrees, 

we multiply the number by f ; and to convert 
Fahrenheit degrees into centigrade degrees we multiply by |. 
In finding the temperature on one scale that corresponds to a 
given temperature on the other scale, it must be remembered 
that the number that expresses the temperature on a Fahren- 
heit scale does not, as it does on a centigrade scale, express 
the number of degrees above melting point. For example. 




52° on a Fahrenheit scale is not 52° above melting point, but 
52° - 32° = 20° above it. 

Hence, to reduce a Fahrenheit reading to a centigrade read- 
ing, first siiMract 32 from the given number, and then multiply 
bij |. Thus, 


To change a centigrade 
reading to a Farhenheit 
reading, first multirply the 
given number by f , and then 
add 32. Thus. 

|C + 32=:F. 

214. Development of the 


Tin melts 233= 

Water boils 100° 
Alcohol boils 78° 

Ether boils 35° 
Ice melts 0° 

Though the invention of Mercury .freeze^8.8° H-37.9° 234.2° H 

the thermometer has been 
ascribed to various scien- 
tific men, it did not assume 

a practical shape until 1620, Aicohoifreezes-j 30.50 H-202.9° h.-.s^h 

at the hands of Drebel, a 
Dutch physician. Halley 

mihstitntPfl mprrnrv for Lowest temperature yet attained 
SUDSUlUieCl merCUiy lOr estimated to be about-2200 

spirit in 1730 ; Reaumur 

of Paris modified the instru- ~2^^° 

ment in 1730, and Fahren- ^^^- ^oi- 

heit of Danzig in 1749 ; Celsius of Upsala, Sweden, improved it in 

1742 by adding the scale now known as centigrade. 1 




^2 ir^ "< 

506° H 878.4° 





215. Self registering, or maximum and mirdmum thermorp.c- 


These are thermometers which enable us to ascertain the highest 
or lowest temperature to which they have been exposed in a given 

1 The Reaumur thermometer is used generally in Germany, the centigrade in 
France, and the Fahrenheit in England, "A prophet is not without honor save in 
his own country." 



interval of time, ordinarily a day. The maximum thermometer 
(the lower one, Fig. 202) is an ordinary mercurial thermometer, 
except that the bore of the tube near the bulb is reduced in such a 
way that while the expansion of the mercury is sufficient to force 
the liquid past the constriction, the cohesion of the liquid is insuffi- 
cient to draw it back again when the temperature falls. To set the 
thermometer it is placed in a vertical position and shaken. This 
causes the mercury to return past the constriction to the bulb, and 
then the instrument indicates the same temperature as that of the 
air. The instrument is then placed in a horizontal position as shown 
in the figure. 

The 7ninimum thermometer (the upper one. Fig. 202) contains 
alcohol, since the freezing point of alcohol is far below that of 
mercury. The bore of the tube contains a little index of metal or 
black glass which moves with a little friction in the tube. It is 
entirely enveloped in the spirit, and the action is as follows : The 

Fig. 202. 

instrument is set by placing it first in an inverted position to allow 
the index to run down the tube to the end of the liquid column, 
and it is then placed in a horizontal position. If the temperature 
rise, the spirit flows past the index without disturbing it. If the 
temperature fall below that at which the instrument was set, the 
capillary action between the spirit and the index is such as to pre- 
clude its leaving the index ; accordingly this is drawn back with the 
spirit, its upper end being always flush with the extremity of the 
liquid column, and ultimately marking the lowest temperature 
reached by the column. 


Requirements of a thermometric substance. 

The sui)stances most commonly employed for thermometric pur- 
poses are mercury, alcohol, and air. The requirements are as 


follows : (1) The substance should be uniformly expansible. In this 
respect the air-thermometer most nearly meets the requirement ; 
but there are great inconveniences attending its use. The expansion 
of mercury is nearly uniform between the melting and boiling points 
on the scale and for a considerable range on each side of them. 
Alcohol, on account of lack of uniformity in its rate of expansion, 
is ill-adapted for this use. (2) A thermometric liquid should have a 
high boiling point. A mercury thermometer will indicate tempera- 
ture as high as about 327° C. (620° F.). The boiling point of alcohol 
is below that of water. (3) The freezing point of the liquid should 
be low. This furnishes the only reason for the use of alcohol. (4) 
The sensibility of the liquid to sudden changes of temperature is 
of importance. On account of the high conductivity of mercury 
and its low specific heat (§ 221), a mercurial thermometer acquires 
thermal equilibrium with the surrounding body more quickly than 
any other liquid thermometer. 

217. Standard thermoTneter. 

As its name indicates, this instrument is used as a standard for 
reference, and for testing from time to time the accuracy of ther- 
mometers used for ordinary observations. 

It is a thermometer which has been compared for every degree 
with an air thermometer, and has a table of corrections accom- 
panied with a certificate of authority from the laboratory where it 
was compared. Owing to the fact that glass after having been 
fused does not immediately return to its normal density, it is neces- 
sary to use for a standard thermometer a tube which has lain 
several years after being filled. The contraction of the bulb causes 
a thermometer to read too high if it was graduated before the 
contraction was completed. The result of this defect in a ther- 
mometer is called the displacement of zero. The graduation of a 
standard thermometer is made on the glass stem, and thus com- 
plication due to difference of dilation of different substances is 

218. Determination of extremely high temperatures. Pyrom- 

Any contrivance for determining temperature above the range 
of a mercurial thermometer (327° C.) is called a pyrometer. There 
are several varieties. Some are constructed on the principle that 



Fig. 203. 

a change of temperature affects the electrical 
resistance of a metal. A hydropyrometer indi- 
cates high temperatures by the number of 
degrees a given mass of water is raised, by 
the immersion in it of a platinum ball of 
known weight after it has acquired the tem- 
perature, for instance, of a furnace or oven 
to be tested. DanielPs pyrometer is one of 
the most useful in practice. The indications 
are obtained from the difference in expan- 
B sion of a platinum bar A (Fig. 203) and a 
tube of black lead B in which the bar is con- 
tained. The index C moves over the scale 
D as the metal rod expands. The degrees 
on this scale are easily converted into those of 
Fahrenheit or centigrade scales. 

Section V. 


219. Distinction between the questions '^ hoiv hot " and " how 
much heat.^^ — The former, like the question '^how sweet/' 
when applied to a solution of sugar, is answered only rela- 
tively. The latter, like the question "how much sugar in 
the solution," is answered quantitively. Sweetness and tem- 
perature are independent of the mass of the body. Quantity 
of sugar depends upon the sweetness and the mass of the 
liquid ; quantity of heat depends upon the temperature and 
the mass of the body. A pint of boiling water is as hot as 
a gallon of the same ; but the latter contains eight times as 
much heat. Temperature depends on the average kinetic energy 
of the molecules. Quantity of heat is the product of the avej^age 
kinetic energy of the molecules multiplied by the number of 

220. Thermal units. — A thermal unit is the quantity of 
heat required to produce a definite effect. The thermal unit 
generally adopted is the ccdorie, which is the quantity of 


heat necessary to raise one kilogram of water from 4° to 5° 
C.^ The thermal unit in the C. G. S. system is the grain-calorie, 
sometimes called the smaller calorie, which is the quantity of 
heat required to raise one gram of water from 4° to 5° C. In 
defining a thermal unit it is necessary to state the tempera- 
tures between which the water is raised, because, although 
the quantity of heat required to raise a given quantity of 
water one degree is very nearly the same at different tempera- 
tures, and in practice is usually regarded as the same, yet the 
quantity required is a very little greater at high temperatures 
than at low temperatures (see § 223). The operation of meas- 
uring heat is called calorimetry. 

221. Heat capacity, specific heat. — The expression heat 
capacity applied to a body refers to the quantity of heat 
necessary to raise the temperature of the body 1°. The 
expression specific heat is applied only to some particular 
substance and refers to the quantity of heat required to raise 
one kilogram of that substance from 4° to 5° C. It is apparent 
that the specific heat of a substance is the heat capacity of 1 unit 
of mass of that substance. 

Experiment 1. — Mix 1 K of water at 0° with 1 K at 20° ; the tempera- 
ture of the mixture becomes 10°.. The heat that leaves 1 K of water 
when it falls from 20° to 10° is just capable of raising 1 K of water from 
0° to 10°. 

Experiment 2. — Take (say) 300 g of sheet lead, make a loose roll of it, 
and suspend it by a thread in boiling water for about five minutes, that 
it may acquire the same temperature (100° C.) as the water. Eemove 
the roll from the hot water, and immerse it as quickly as possible in 
300 g of water at 0°, and introduce the bulb of a thermometer. Note the 
temperature of the water when it ceases to rise, which will be found to 

1 Authorities do not agree on the temperattire limits for this unit. Some German 
authorities give 15° to 16'' C. Regnault chose 0° to 1° C, and this has been quite 
generally adopted in scientific treatises. There seem, however, to be good reasons 
for a departure from this custom, and we have chosen the limits proposed by 
Glazebrook in his recent treatise on Heat, viz., 4° to 5° C. 


be about 3° (accurately 3.3° +). The lead cools very much more than the 
water warms. The temperature of lead falls about 33° for every degree 
an equal mass of water is warmed. 

From the first experiment we infer that a body in cooling 
a certain number of degrees gives to surrounding bodies as 
much heat as it takes to raise its temperature the same number 
of degrees. From the second experiment we learn that the 
quantity of heat that raises 1 K of lead from 3.3° + to 100°, 
when transferred to water, can raise 1 K of water only from 
0° to 3.3°. Hence we conclude that equal quantities of heat, 
applied to equal masses of different substances, raise their 
temperatures unequally. 

If equal masses of mercury, alcohol, and water receive 
equal quaDtities of heat, the mercurjrwill rise 30°, and the 
alcohol nearly 2°, for every degree the water rises. From 
this we infer that to raise equal masses of each of these 
substances 1° requires 30 times as much heat for the water 
as for the mercury, and twice as much as for the alcohol. 
Since a given quantity of heat affects the temperature of a 
given mass of water less than that of an equal mass of mer- 
cury or alcohol, water is said to have greater specific heat 
than these substances. It is also apparent that a given mass 
of water in cooling imparts to surrounding bodies more heat 
than the same masses of mercury and alcohol would impart 
in cooling the same number of degrees, in proportion to its 
greater specific heat. 

222. Method of measuring specific heat. 

There are at least four methods practiced, only one of which, the 
"method of mixtures," will be considered. A known mass m (in 
kilograms) of the substance of which the specific heat is required is 
taken, as in Experiment 2, and heated to a known temperature t-^ 
(C) ; then it is mixed with (or immersed in) a known mass of water 
m2 at a lower temperature ^2, and as soon as thermal equilibrium is 


established throughout, the temperature of the mixture t is taken. 
Let s represent the specific heat of the substance sought. Then the 
quantity of heat lost by the substance is m X s (^i — t) calories ; 
while that gained by the water is m^ {t — t^) calories. Now if no 
heat be lost during the operation, ?n X s (ti — t) = 1112 [t — ^2), whence 

^ ~ _ V • For example, taking the quantities obtained in the 

experiment above, we find for lead (300 g=^=.3 K) s = '' )' . r-^ 

.»j (100 — 0.0) 
= .034 calorie. 

223. Specific heat of the same substances at different tem- 
peratures and in the three states of matter . — The specific heat 
of solids and liquids usually increases slightly with the tem- 
perature, and diminishes with increase of density. The spe- 
cific heat of water at 0°, 40°, and 80° is respectively 1, 1.003, 
and 1.0089 calories. Substances in the liquid state usually 
have a higher specific heat than in the solid or gaseous state. 
Thus, water has nearly double the specific heat of ice, and a 
little more than double the specific heat of steam. 

The mean specific heat of a substance between 0° and t° is the 
average quantity of heat {e.g. of calories) per degree required in 
heating a unit mass of the body from 0° to t°. Let h be the total 
number of thermal units required to heat the unit mass of the 
substance from 0° to ^°, then the mean specific heat s between 0° 
and t° is expressed by the formula 


' = !■ 

The specific heat of any perfect gas measured by its mass is 
independent of temperature and density ; for an imperfect gas 
(vapor), it increases with the temperature, and diminishes with 
increase of density. The specific heat of all perfect gases measured 
by volume depends on the number of atoms in the molecule, being 
proportional to that number. Thus the specific heats of all diatomic 
perfect gases are nearly the same ; that of a triatomic gas would be 
to these as 3:2, etc. 


Reference Tables. 
Table of mean specific heat between 0° C. and 100° C. 

Copper 095 

Sulphur 2026 

Glass 1770 

Iron ....... .113 

Mercury 033 

Lead 031 

Specific heat of the same substance in different states. 

Solid Liquid Gaseous 

Water 504 . . . 1.000 . . . .480 

Bromine 083 . . . .106 . . . .055 

Lead 031 . . . .040 

Alcohol 55-.77 ... .45 

224. Great ccq^acity of water for heat. — Water requires 
more lieat to warm it, and gives out more in cooling through 
a given range of temperature, than any other substance except 
hydrogen. The quantity of heat that raises a kilogram of 
water from 0° to 100° C. would raise a kilogram of iron from 
0° to 800° or 900° C, or above a red heat. Conversely, a 
kilogram of water in cooling from 100° to 0° C. gives out as 
much heat as a kilogram of iron in cooling from about 900° 
to 0° C. 

"The vast influence which the ocean must exert as a 
moderator of climate here suggests itself. The heat of 
summer is stored up in the ocean, and slowly given out 
during the winter. This is one cause of the absence of 
extremes in an island climate." 

The high specific heat of water is utilized in heating 
buildings by hot water. 

225. Relation between specific heat and atomic mass. 

The heat energy of a molecule of hydrogen is equal to that of a 
molecule of oxygen at the same temperature ; and a mass of 
hydrogen contains sixteen times as many molecules as an equal 
mass of oxygen. Hence a given mass of hydrogen possesses sixteen 


times as much heat energy as an equal mass of oxygen at the same 
temperature. Therefore, to produce a given rise in the temperature 
of a mass of hydrogen, sixteen times as much heat is required as 
for an equal mass of oxygen ; hence the specific heat or thermal 
capacity of hydrogen is sixteen times that of oxygen. 

226. Specific heat of elementary gases varies inversely as 
their atomic masses ; or, the product of the specific heat and 
atomic mass is eonstant. 

When the molecules are constrained by cohesion, as in liquids 
and solids, a part of the heat applied to a body is spent in raising 
its temperature and a part in doing internal work in overcoming 
cohesion between the molecules of the body and in forcing them to 
take up new positions. The greater the portion of heat consumed, 
i.e. converted into potential energy, in doing internal work, the less 
there is left to raise its temperature. Hence the law as given for 
gases holds only approximately for liquids and solids. 

Section VI. 

effects of heat. expansion. 

Having learned something of the nature of heat and the 
methods by which it is measured, w^e will next direct our 
attention to some effects it produces, viz. expansion and 
change of state. The first, as we have learned, furnishes a 
means of measuring temperature and leads to a fuller study 
of gases than has yet been made. Under the second effect 
we study liquefaction and vaporization. 

227. Experiments illustrating expansion of solids, liquids, 
and gases. 

Experiment 1. — Take two brass tubes, one of a size that will permit 
it just to enter the bore of the other. Heat the smaller tube ; it will not 
in its expanded state enter the other. Thrust the heated tube into cold 
water ; its temperature falls, and it now enters the bore of the other tube. . 
"Heat expands," but "cold" does not "contract." Cohesion, when a 



diminution of heat (which acts as a repellent force) permits, causes a 
solid or liquid body to contract. Cold is a term of negation signifying 
merely a greater or less deficiency of heat ; it is not an entity, hence it 
cannot be the direct cause of any phenomenon. 

Experiment 2. — Fig. 204 represents a thin brass plate and an iron 
plate of the same dimensions riveted together so as to form what is called 

a compound bar. Place the bar edgewise 

i"" ' ""'I in a flame, dividing the flame in halves 

(one half on each side of the bar) so 
that both metals may be equally heated. 
The bar, which was at first straight, is now bent, owing to the unequal 
expansion of the two metals on receiving equal increments of temperature. 

Fig. 204. 

Fig. 205. 

Fig. 206. 

When heated above the normal temperature, the brass, which is more 
expansible, will be on the convex side ; when cooled below the normal 
temperature, it will be on the concave side, since it contracts more 
rapidly than iron. 

Metallic thermometers now m common use (Fig. 205) are con- 
structed on this principle. They contain a compound ribbon of 
metal (Fig. 206) wound into a spiral, one end of which a is fixed 


SO as to be immovable, while the other is attached to a contrivance 
for multiplying motion which moves the index. With a rise in tem- 
perature, the more expansible metal on the outside produces an 
increase of curvature, which causes the spiral to wind up closer. 
This motion is communicated to the index, which points on the 
dial to the corresponding temperature. With a fall of temperature 
the action is reversed. 

Advantage is taken of this principle also in the construction of 
balance wheels of chronometers. The rate of vibration of a chro- 
nometer balance wheel depends upon its mass 
and the distance of its circumference from the 
center. The parts BC and FG (Fig. 207) are 
made up of a compound strip, the more expan- 
sible metal being on the outside. As the tem- 
perature rises the radii AA expand, and the 

chronometer would lose time, but the heat 

' Fig. 207. 

causes the strip BC and FG lo curve inwards. 

The masses D and D' are thus brought nearer the center, and this 

compensates for the expansion of A. 

Experiment 3. — Fit stoppers tightly in the necks of two similar thin 
glass flasks (or test-tubes), and through each stopper pass a glass tube 
about 60 cm long. The flasks must be as nearly alike as possible. Fill 
one flask with alcohol and the other with water, and crowd in the stop- 
pers so as to force the liquids in the tubes a little way above the corks. 
Set the two flasks into a basin of hot water, and note that, at the instant 
the flasks enter the hot water, the liquids sink a little in the tubes, but 
quickly begin to rise, until, perhaps, they reach the top of the tubes and 
run over. 

When the flasks first enter the hot water they expand, and thereby 
their capacities are increased ; meantime the heat has not reached the 
liquids to cause them to expand, consequently the liquids sink momen- 
tarily to accommodate themselves to the enlarged vessel. Soon the heat 
reaches the liquids, and they begin to expand, as shown by their rise in 
the tubes. The alcohol rises faster than the water. Different substances, 
in both the solid and the liquid states, expand unequally on experiencing 
equal changes of temperature. 

Experiment 4. — Take a dry flask like that used in Exp. 3, insert 
the end of the tube in a bottle of colored water (Fig. 208), and apply 
heat to the flask ; the enclosed air expands and comes out through the 
liquid in bubbles. After a few minutes withdraw the heat, keeping the 



end of the tube in the liquid ; as the air left in the flask cools, its pressure 
decreases, and the water is forced by atmospheric pressure up the tube 
into the flask, and partially fills it. 

Experiment 5. — Partly fill a foot-ball with cold 
air, close the orifice, and place it near a fire. The 
air will expand and distend the ball. 

228. Expansion - coefficients. — The ex- 
pansion which attends a rise of tempera- 
ture depends not only upon the size of the 
body, and upon the number of temperature 
degrees through which it is heated, but 
upon a quantity peculiar to the substance 
itself called its expansion-coefficient. This 
term is applied to the increase of unit-length 
per degree rise of temperature. 

Suppose that a rod of length I at 0° C. be 
heated through t degrees, so that its length 

becomes l^ \ then, representing the linear expansion-coefficient 

by c, we have 


Fig. 208. 


whence Z^ ^ Z (1 -|- ct>). 

The expression 1 -\- ct^ called the expansion- factor., is evi- 
dently the ratio of the final to the original length. Hence 
Zi == Z (1 + ct) \ that is, multiplying the length of a solid at 
0° C. by the expansion factor gives its length at t degrees 
above zero. Conversely, dividing its length at t° by the 
expansion factor gives its length at 0°. 

Table of Mean Coefficients of Linear Expansion between 0° 

AND 100° C. 

Glass ...... 0.0000085 

Platinum 0.0000085 

Steel 0.000012 

Wrought iron . . . 0.000012 

Cast iron ..... 0.000011 

Gold . 0.000015 

Copper ...... 0.000017 

Brass 0.000019 

Silver 0.000019 

Tin 0.000022 

Lead 0.000029 

Zinc ....... 0.000029 


In the expansion of fluids we have to do only with increase 
of volume, called volume or cuhical expansion. A volume- 
expansion-coefficient is the increase of unit volume per degree 
rise of temperature. This is approximately 3 c, or three times 
the linear expansion-coefficient, and may be taken as such for 
most practical purposes. Likewise, the surface or superficial 
expansion-coefficient is approximately 2 c. 

Not only do the expansion-coefficients of liquids and solids 
vary with the substance, but the coefficient for the same 
substance varies with the temperature, being greater at high 
than at low temperatures. Hence, in giving the expansion- 
coefficient of any substance it is customary to give the mean 
coefficient through some definite range of temperature, as 
from 0° to 100° C. 

229. Force exerted in expansion and contraction. — The force 
which may be exerted by bodies in expanding or contracting 
may be very great, as shown by the following rough calcula- 
tion : If an iron bar, 1 sq. in. in section, be raised from 0° C. 
(melting point of ice) to 500° C. (a dull red heat), its length, 
if allowed to expand freely, will be increased from 1 to 1.006, 
its expansion-coefficient being about .000012. Now, a force 
of about 90 tons is required to stretch a bar of iron of 
1 sq. in. section this amount, and this is very nearly the force 
that would be necessary to prevent expansion caused by the 
heat. It would require an equal force to prevent contraction 
(caused by what ?) if the bar be cooled at 500°. 

Boiler plates are riveted with red-hot rivets, which, on 
cooling, draw the plates together so as to form very tight 
joints. Tires are fitted on carriage-wheels when hot, and, on 
cooling, grip them with very great force. 

It is to be observed that while the force exerted in expan- 
sion and contraction is great, the distance through which it 
acts is very small, and hence the quantity of work performed 
is not very great. 


230. Anomalous expansion and contraction. — Water pre- 
sents a partial exception to the general rule that matter 
expands on receiving heat and contracts on losing it. If a 
quantity of water at 0° C, or 32° F., be heated, it contracts 
as its temperature rises, until it reaches 4° C, or about 39° ¥., 
when its volume is least, and therefore it has its maximum 
density. If heated beyond this temperature it expands, and 
at about 8° C. its volume is the same as at 0°. On cooling, 
water reaches its maximum density at 4° C, and expands as 
the temperature falls below that point. 

Water is said to have a negative expansion-coefficient be- 
tween 0° and 4° C, or between 32° and 39.2° F. A few other 
substances, such as India rubber and iodide of lead, contract 
when heated, and have, therefore, negative coefficients. 

Section VII. 


231. Kinetic theory of matter. — In the case of solids the 
molecules are thought to move in curved orbits the centers of 
which are fixed. In liquids the orbits are curved; but, as 
shown in the phenomena of diffusion (p. 138), the molecules 
have, besides the oscillating motion, a motion of translation. 
The theory that the molecules composing all bodies of matter 
are in perpetual relative motion is called the kinetic theory of 
matter. This theory claims that in gases the molecules are so 
far separated from one another that their motions are not 
generally influenced by molecular attractions. Hence, in 
accordance with the first law of motion, the molecules of 
gases move in straight lines and with uniform velocity until 
they collide with one another or strike against the walls of 
the containing vessel, when, in consequence of their elas- 


ticity, they at once rebound and start on a new path. We 
may picture to ourselves what is going on in a body of calm 
air, for instance, by observing a swarm of bees in which 
every individual bee is flying with great velocity, first in one 
direction and then in another, while the swarm either remains 
at rest or sails slowly through the air. 

232. P7'essure of a gas due to the kinetic energy of its 
molecules. — Consider, then, what a molecular storm must be 
raging about us, and how it must beat against us and against 
every exposed surface. According to the kinetic theory, the 
pressure of a gas (or its expansive force as it is sometimes 
called) is entirely due to the striking of the molecules against 
the surfaces on which the gas is said to press, the impulses 
following one another in such rapid succession that the effect 
produced cannot be distinguished from constant pressure.^ 
Upon the kinetic energy of these blows, and upon the number 
of blows per second, must depend the amount of pressure. 
But we have learned that on the energy of the individual 
molecules depends that condition of a gas called its tempera- 
ture; so, it is apparent, as stated above, that the pressure of a 
given quantity of gas varies ivith its temperature. Again, as 
at the same temperature the number of blows per second 
must depend upon the number of molecules in the unit of 
space, it is apparent that the j^ressure vai^ies with the density. 

' ' If the rarefaction of air can be carried so far that only one 
particle out of every million is left in the space exhausted, the mean 
path of the particles would then be about 4 inches. In our atmos- 
phere at a hight of 210 miles, the particles are relatively so few 

1 The following estimates made (by Maxwell, using a proposition formulated by 
Clausius) for hydrogen molecules at 0°C., and under a pressure of 760 mm, may 
prove interesting : 

Mean velocity, 6100 feet per second. 

Mean path without collision, 38 ten-millionths of an inch. 

Collisions, 17,750 millions per second. 

Mass, 216,000 million million million in 1 gram, 

Nuniber, 19 million million million fill 1 cubic centimeter. 



that each particle might travel through a uniform atmosphere of 
that density for sixty million miles without entering into collision." 
— Daniell. 

233. Expansion and expansive force of gases. 

The effect of a change of temperature upon a gas may be meas- 
ured by noting the change in its volume when the pressure upon 
it is constant, or the change in its pressure when its volume is 
unchanged. Conversely, the changes in volume or pressure of a 

Fig. 209. 

gas may be made to indicate changes in temperature. On this 
principle the so-called air-thermometer is constructed. 

The relation between pressure and temperature of air kept at a 
constant volume may be found by means of an apparatus like that 
represented in Eig. 209. A bulb &, whose capacity at 0° and 
100° C. is known, is filled with dry air. The capillary tube leading 
from the bulb is connected to a tube T, which is connected with 


another tube T' open to the atmosphere. The lower ends of T and 
T^ dip into a reservoir of mercury E. The bulb b is first sur- 
rounded by melting ice, and by means of the screw S the mercury 
is forced to the hight h in the tube T, and the difference in level 
between {h and h') the surfaces of mercury in the two tubes is 
ascertained. By adding this to the hight of the barometer at the 
same moment, the total pressure to which the air in the bulb is 
subjected at the temperature (0°C.) of melting ice is ascertained. 
The bulb is next introduced into an apparatus for boiling water, as 
shown in the figure, and surrounded with steam from the boiling 
water. By means of the screw S the mercury is again forced to 
the same hight h as before in the tube T. But since the pressure of 
air increases with the temperature, the mercury will now be much 
higher in tube T' and higher in proportion to the increased pressure. 
By this means Regnault ascertained that the pressure of dry air 
confined to the same volume is about 1.367 times greater at 100° 
than at 0° C. ; in other words, the increase of pressure for 100° is 
.367, or (.367 -MOO = ) .00367 per degree. 

By a slight modification of this instrument and a variation in the 
method of using it, Regnault ascertained that when air is allowed 
to dilate with increase of temperature while the pressure remains 
constant, the volume at 100° is increased by .367 its volume at 0°. 
Dividing this number by 100, we obtain .00367 for the expansion 
coefficient of air between 0° and 100° C. 

It is found that the expansion-coefficient of all gases is approxi- 
mately the same as long as they remain true gases, but as they 
approach the vaporous state the coefficient changes rapidly. 

234. Absolute zei'o. — The zeros on the thermometrio scales 
which we have hitherto considered are provisional, arbitrary. 
Absolute zero is the temperature corresponding to total 
absence of heat. At the absolute zero the molecules must be 
supposed to be at rest. At this temperature gases (if they 
may be called such) exert no pressure, and occupy no space 
save that which their molecules take up when closely packed 
together. The point of absolute zero is independent of the 
conventions of man. It is a point of absolute cold or absence 
of heat, beyond which no cooling is conceivable. 

Now it has been found (§ 233) that the pressure in air 


increases or diminislies by .00367 = (about) gi^ of its pressure 
at 0° for each centigrade degree of rise or fall of tempera- 
ture, the volume being maintained constant. If air were a 
perfect gas, and could be cooled down in this way to — 273° 
C. ( — 459.4° F.), it would exert no pressure. The reason it 
would exert no pressure is that its particles possess no kinetic 
energy, no motion. This is assumed, therefore, to be the 
absolute zero of temperature. 

235. Thermo-dynamic definition of temperature. 

In this system, temperature, i, is defined by the equation E = kt, 
in which E is the average kinetic energy per molecule of a perfect 
gas which has that temperature, and fc is a constant. This is called 
the thermo-dynamic definition of temperature. 

236. Absolute temperature. — Absolute temperature is that 
reckoned from the absolute zero, or — 273° C. Temperatures 
Tneasured from absolute zero are proportional to the pressure of 
a theoretically perfect gas of constant volume or density ; this 
statement is merely a convenient expression of the laws of a 
perfect gas (§ 237). 

The absolute temperature (based on the above theory) of 
any body is found by adding 273 to its temperature as 
indicated by a centigrade thermometer, or 459.4 to its tem- 
perature as indicated by a Fahrenheit thermometer. The 
comparative scale given on p. 257 will make this clear. 

237. Laws of gaseous masses. — It follows, from the above 
discussion, (1) that the volume of a given mass of gas at con- 
stant pressure is proportional to its absolute temperature; i.e. at 

V (volume of a sriven mass of eras) 

constant pressure — ^ — T^ry-y — . , , r^ — - remains 

T (absolute temperature) 

constant. This is called the Law of Charles. 

(2) The pressure of a given 7nass of gas luhose volume is kept 
constant is j^roportional to its absolute temperature. 

Boyle's law states that (3) at a constant temperature the 
volume of a given mass of gas is iiiversely proportional to its 


pressitre; i.e. the iiroduct of its jpressure and its volume is con- 
stant. NoWj when both the pressure and the volume vary at 
the same time, it may be shown that (4) the product of the 
pressure and the volume of a given mass of gas is proportional 
to its absolute temperature. A gas is said to be perfect, when 
it perfectly obeys these laws. 

We may also state this law as follows : the product of the 
pressure and volume of a given mass of gas divided by its 
absolute temperature is a constant quantity, or 

in which P = pressure, V^ volume, T= absolute temperature 
of a given mass of gas, and 6' = a constant quantity, the value 
of which depends on the gas in question. 


1. Find, in both centigrade and Fahrenheit degrees, the absolute tem- 
peratures at which mercury boils and freezes. 

2. At 0° C. the volume of a certain mass of gas under a constant 
pressure is 500 cc ; a. What will be its volume if its temperature be 
raised to 75° C? 6. What will be its volume if its temperature become 
— 20°C.? 

3. If the volume of a mass of gas at 20° C. be 200 cc, what will be its 
volume at 30° C? Solution: 20° C. is equivalent to (20 + 273) 293 abs. 
temp. ; then 293 : 303 : : 200 : 206.8 cc. Ans. 

4. To what volume will a liter of gas contract if cooled from 30° C. to 

5. One liter of gas under a pressure of one atmosphere will have what 
volume, if the pressure be reduced to 900 g per square centimeter, while 
the temperature remains constant ? 

6. The volume of a certain mass of air at a temperature of 17° C, 
under a pressure of 800 g per square centimeter, is 500 cc ; what will be 
its volume at a temperature of 27° C, under a pressure of 1200 g per 
square centimeter? Solution: 17° C. is equivalent to 290° abs. temp. ; 
27° C. is equivalent to 300° abs. temp. Then 290 : 300 : : 500 X 800 : x X 
1200. Whence x = 344.8 cc. Ayis. 

7. If the volume of a mass of gas under a pressure of 1 K per square 


centimeter at a temperature of 0° C. be 1 liter, at what temperature will 
its volume be reduced to 1 cc under a pressure of 200 K per square centi- 
meter ? Ans. : 54.6° abs. temp., or — 218.4° C. 

8. Find the temperatures on the absolute scale at which the substances 
named on p. 278 melt. 

9. If a cubic foot of coal-gas at 32° F. , when the barometer is at 30 in. , 
has a mass of 2V ^^-i what will be the mass of an equal volume at 68° F., 
when the barometer is at 29 in. ? 

10. Explain the following statement : "To compare absolute tem- 
peratures, we may seal up a mercurial barometer in a tube, or an aneroid 
barometer in a preserving jar." Whiting. 

Section VIII. 


238. Change of pi^operties in solids attending change of tem- 
perature. — " Every known property of a piece of matter, 
except its mass, varies with variation of temperature." Inas- 
much as heat tends to weaken cohesion, the rigidity and 
tenacity of solids are generally lessened, and their plasticity 
is increased, by the addition of heat. 

239. Fusioji. — As previously stated, whether a given sub- 
stance exist in a solid, liquid, or gaseous state depends upon 
the temperature and the pressure it is under. Solids exposed 
to heat liquefy or fuse, unless previously decomposed. Some, 
like ice and tin, change their state abruptly ; others, like 
glass and wrought iron, become plastic prior to liquefaction. 
The temperature at which a substance melts is called its 
fusion-point. The fusion-points of different substances vary 
greatly : that of alcohol ( — 130.5° C.) and that of iridium 
(1950° C.) may be taken as extreme examples. 

Experiment and experience teach that (1) the melting or 
solidifying point (they are approximately the same for the same 
substance) m.ay vary ividely for different substances, but for the 
same substance it is invariable when under the same pressure. 


(2) The temioeratuve of a solid or liquid remains constant at 
the nielting-])oint from the moment that melting or solidifteation 
begins until it ceases. 

Exjperiment 1. — Put a lump of ice as large as your two fists into 
"boiling water ; when it is reduced to about J its original size, skim it out. 
Wipe the lump, and place one hand on it and the other on a lump to 
which heat has not been applied ; you will not perceive any difference in 
their temperatures. Under ordinary pressure ice cannot be made warmer 
than 0° C. 

240. Change of volume during fusion or solidification. 
During these changes of state there is usually a change of volume. 

Generally solids expand on melting, so that the volume is increased 
and the density is diminished by fusion. There are, however, 
certain important exceptions to this rule. Ice and type-metal, for 
example, float in the denser liquids of these substances respectively. 
Such metals as expand on solidifying are especially adapted for 
casting, since they expand and fill every portion of the mould, and 
the resulting impression is sharp and clear. 

241. Effect of pressure on the fusion point. 

In solids whose volume increases during fusion, the fusion is more 
difficult when the pressure is increased, since not only is heat 
required to change the state of the body, but an additional quantity 
is required to do additional external work against the additional 
pressure. In solids whose volume decreases during fusion, an 
increase of pressure facilitates the fusion, and the temperature is 
less as the pressure is greater. Briefly, then, pressure lowers the 
fusion point of substances that expand on solidifying, and raises the 
fusion point of those that contract. 

24:2. Regelation. 

Experiment 2. — Place in contact the smooth surfaces (wiped 
dry) of two pieces of ice ; press them together for a few seconds ; 
remove the pressure, and they will be found to be firmly frozen 

This phenomenon is called regelation, and is simply the con- 
sequence of lowering the fusion point by pressure. The fusion 
point being lowered, the ice at the surfaces of contact is melted, but 
as the water produced is below the normal fusion point of water, it 
freezes again as soon as the pressure is removed. 



Table of Edsion Points. 

Alcohol -130.5° C. 

Mercury . . 
Sulphuric acid 
Ice . . . . 
Phosphorus . 
Sulphur . . 
Tin ... 
Lead . . . 

. -38.8° 

. -34.4° 




about 233° 

" 334° 

Zinc . 

Silver . . 
Gold . . . 
Cast iron . 
Wrought iron 
Platinum . 
Iridium (the most 
infusible metal) 


425° C. 

. . 954° 

. 1200° 



. 1775° 

. 1950° 

243. Heat of fusion. — The temperature of ice remains 
constant while melting, and generally heat imparted to a 
melting body affects its temperature very little if any. 
Furthermore, ice and other solids are not instantly converted 
into liquids on reaching the fusion point, but absorb a quan- 
tity of heat proportionate to their mass before fusion is 
accomplished. Inasmuch as none of the heat applied during 
melting raises the temperature of the body, the question 
arises ivhat becomes of the heat applied to the body? The 
thermo-dynamical theory furnishes the only satisfactory 
answer to this important question. The answer is, about all 
the heat applied to a body during fusion is consumed in doing 
internal ivot'k, as it is called. The molecules that were held 
firmly in their places by molecular forces are, during fusion, 
moved from their places, and so work is done against these 
forces. Heat, the energy of motion, performs this work, and 
is thereby converted into pMential energy, the energy of 
position, — energy of the same kind as that of a raised 
weight. The heat which disappears in melting is called the 
heat of fusion. 

If it require a large quantity of heat and a long time to 
effect the fusion of a body, it must be inferred that the 
amount of work done is proportionately great. Fortunate is 
it that it does require much heat to melt moderately small 


masses of ice and snow, else on a single warm day in winter 
all the ice and snow would melt, creating most destructive 

244. Measurejneyit of the heat of fusion. — Let it be required 
to find approximately the quantity of heat that disappears 
during the melting of one kilogram of ice. This quantity is 
most readily determined by the method of mixtures. 

Experiment 3. — Weigh out 200 g of dry ice chips (dry them with a 
towel), whose temperature in a room of ordinary temperature may be 
safely assumed to be 0° C. Weigh out 200 g of boiling water, whose 
temperature we assume to be 100° C. Pour the hot water upon the ice, 
and stir it imtil the ice is all melted. Test the temperature of the resulting 

Suppose its temperature is found to be 10° C. It is evident that the 
temperature of the hot water in falling from 100° to 90° would yield 
sufficient heat to raise an equal mass of water from 0° to 10° C. Hence 
it is clear that the heat which the water at 90° yields in falling from 90° 
to 10° — a fall of 80° — in some manner disappears. At this rate had you 
used 1 k of ice and 1 k of hot water, the amount of heat lost would be 80 
calories. Careful experiments, in which suitable allowances i are made 
for loss or gain of heat by radiation, conduction, absorption by the calo- 
rimeter, etc. , have determined that 80 calories of heat are consumed in 
melting 1 kilogram of ice. 

Table ^ of Heats of Fusion of Substances under the Pressure 
OF ONE Atmosphere. 

Calories Calories Calories 

per kilogram. per kilogram. per kilogram. 

Water .... 80.0 Silver . . 24.7 Sulphur . . 9.4 

Sodium nitrate . 63.0 Tin . . . 14.25 Lead . . . 5.4 

Potassium nitrate 47.4 Bismuth . . 12.6 Phosphorus . 5.0 

Zinc. .... 28.1 Iodine . . 11.7 Mercury. . 2.82 

According to this table it is to be understood, for example, 

1 As this is not a manual of manipulation, the cumbersome details relating to 
cautions against errors which must be observed in order to secure even approximately 
accurate results are omitted. The student will find ample directions in almost all of 
the many laboratory manuals extant. 

2 Barker, 


that one kilogram of ice at 0° "ander tlie pressure of one 
atmosphere, while changing to a liquid, absorbs without any 
change of temperature 80 calories of heat, or as much heat as 
is required to raise 80 kilograms of water from 0° to 1° C. 
It will be seen that water possesses the greatest latent heat 
of fusion. 

245. Transformation of heat reversible. — As stated at the 
beginning of this chapter, work is transformable into heat, 
and, as stated on p. 87, potential energy is transformed into 
kinetic energy '^ by the return of the molecules to their orig- 
inal positions ; " so when water freezes or any liquid is 
re-solidified, the potential energy (latent heat of fusion) reap- 
pears as heat. 

Water in freezing undergoes no change of temperature, 
hence, if heat be developed during the operation, it must 
become diffused or must be " given off " in order to allow the 
freezing to go on. As the diffusion is necessarily slow, so 
freezing must be slow ; and this slow development of heat 
and its immediate dispersion accounts for the fact that we 
are seldom made conscious of the development of heat during 

Farmers sometimes turn to practical use this well-known 
phenomenon. Anticipating a cold night, they carry tubs of 
water into cellars to be frozen. The heat generated thereby, 
although of a low temperature, is sufiicient to protect vege- 
tables which freeze at a lower temperature than water. 

Heat disappears in the process of melting ice ; and, para- 
doxical as it may seem, heat is generated by freezing water. 
By freezing one kilogram of water 80 calories of heat are 


Section IX. 


246. Evaporation ; ebullition. — The process of converting 
a liquid into a vapor is called vaporization. A comparatively 
slow vaporization which takes place only at the exposed sur- 
face of a liquid is called evap)oration. A rapid process which 
may take place throughout the liquid, but usually is most 
rapid at the point where heat is applied, is called boiling or 

247. Kinetic theory of evaporation and condensation. — 
According to this theory, some of the molecules in any liquid 
move faster than others. Those at the surface which have 
great velocities, if the direction of their motion be from the 
liquid, will break away from the forces that are able to retain 
the molecules moving more slowly, and will fly about as 
vapor in the space outside the liquid. This is evaporation. 
At the same time molecules of the vapor striking the liquid 
may plunge into it and become entangled in it, and thus 
there is a return to the liquid state. This is condensation. 
The number of molecules which passes from the liquid to the 
vapor increases with increase of temperature of the liquid. 
The number which passes from the vapor to the liquid de- 
pends upon the density of the vapor as well as its tempera- 
ture. When the density of the vapor increases to such an 
extent that as many molecules are condensed as are evaporated, 
then the vapor is said to have its maximum density for that 
temperature, or to be saturated. The evaporation then ap- 
pears to cease, because the proportions of liquid and vapor 
remain unchanged. Liquids which evaporate readily are 
called volatile liquids in distinction from those which do not, 
and which are called fixed liquids. 

248. Boiling point. — In evaporation, molecules fly from 



the surface of the liquid and mingle with the particles of the 
air and drive only a certain small portion of them away. In 
boiling, the molecules which fly away from the surface drive 
all the air particles away a certain distance. Hence the 
vapor of a boiling liquid must exert a pressure at least 
as great as the atmospheric pressure. The greater the 
external pressure to be overcome, the greater must be the 
energy, i.e. the higher the temperature, of the vapor. When 
the saturated vapor of a liquid exerts a pressure equal to that 
of the atmosphere, the liquid begins to boil, and the tempera- 
ture at which this occurs is called the normal boiling point of 
that liquid. 

Experiment 1. — Half fill a glass flask with water. Boil the water 
over a Bunsen burner ; the steam will drive the air from the flask. With- 
draw the burner, quickly cork the flask very tightly, and plunge the flask 

into cold water, or invert the flask and 
pour cold water upon the part contain- 
ing steam, as in Fig. 210 ; the water in 
the flask, though cooled several degrees 
below the usual boiling point, boils again 
violently. The application of cold wa- 
ter to the flask condenses some of the 
steam, and diminishes the pressure of 
the rest, so that the pressure upon the 
water is diminished, and the water boils 
at a reduced temperature. 

If hot water be poured upon 
the flask, the water ceases to boil. 
Under the receiver of an air-pump, 
water may be made to boil at 
any temperature between 0° and 
100° C. ; indeed, if exhaustion be 
carried far enough, boiling and freezing may be going on at the 
same time. When high temperature is objectionable, appara- 
tus is contrived for boiling and evaporating in a vacuum ; as, 

Fig. 210. 


for instance, in the vacuum pans used in sugar refineries. As 
water boils more easily under diminished pressure, so it boils 
with more difficulty when the pressure is increased ; and the 
temperature to which water may be raised under the pressure 
of its own steam is limited only by the strength of the vessel 
containing it. Vessels adapted to resisting steam pressure, 
called digesters, are often employed to effect a complete pene- 
tration of water into solid and hard substances. By this 
means gelatine is extracted from the interior of bones. In 
the boiler of a locomotive, where the pressure is sometimes 
150 lbs. above >the atmos]3here, the boiling point rises to about 
185° C. (365° F.). 

Experiment 2. — Place a test tube (Fig. 211) half filled with ether in a 
beaker containing water at a temperature of 60° C. Although the tem- 
perature of the water is 40° below its boiling point, it very 
quickly raises the temperature of the ether sufficiently to 
cause it to boil violently. Introduce a chemical thermom- 
eter into the test tube, and ascertain the boiling point of 

Experiment 3. — Li a beaker half full of distilled water 
suspend a thermometer so that the bulb will be covered by 
the water and yet be at least two inches above the bottom -p^^ 211 
of the beaker. Apply heat to the beaker, and observe any 
changes of temperature which may occur, both before and after boiling 

Experiment 4- — Dissolve table-salt in water, and you may raise its 
boiling point till it reaches 108° C. With saltpetre it may reach 115° C. 

It is found that (1) /or a given pressure (for example, that 
of the atmosphere at 760 mm) every liquid has a definite boil- 
ing point ; (2) this hoiliiig point remains constant after boiling 
has begun; (3) salts dissolved in liquids raise their boiling 
points, but do not ajfect the temperature of the escaping vapor. 
The latter result is obviously due to the additional work 
required to be performed by the heat in order to overcome 
the increased cohesion due to the salt in solution. 



Boiling Points under 

Sulphurous anhydride . — 8° C. 

Ether 34.89° 

Carbon disulphide . . 46.8° 

Bromine 63° 

Wood-spirit .... 65.50° 

Alcohol 78.39° 

A Pressure of 760 mm 
Benzole . . 
Water . . . 
Acetic acid . 
Butyric acid . 
Sulphuric acid 
Mercury . . . 

80.44° C. 


Boiling Points of Water at Different Pressures 
680 mm 


96.9° C. 












100° c. 



The boiling point of water varies with the altitude of 

places, in consequence of the change in atmospheric pressure. 

Eoughly speaking, a difference of altitude of 533 ft. causes a 

variation of 1° F. in the boiling point. The measurement of 

hights by means of the boiling point is called hy^jsoinetry. 

A hypsometer is simply a convenient portable apparatus for 

boiling water, provided witli a thermometer sensitive to (say) 


Boiling Points of Water at Different Altitudes. 

Above the 

Quito + 9,500 ft. 

Mont Blanc . . . 15,650 " 

Mt. Washington . . 6,290" 

Boston " 

Dead Sea (below) . - 1,316 " 

249. Vaporization of solids. — The boiling point of a sub- 
stance under ordinary atmospheric pressure may be below its 
fusion point; if so, the solid changes directly into a vapor 
without passing througli the usual intermediate liquid state. 
For example, the maximum vapor pressure of carbon dioxide 

Mean higlit of 

21.53 in. . 

. . 91° C. 

16.90 " . 

. . 86° 

22.90 " . 

. . 94° 

30.00 " . 

. . 100° 

31.50 " . 

. . 101° 



at its fusion point (—66° C.) is three atmosplieres ; under 
any less pressure it vaporizes, but never melts. Ice cannot be 
melted under less than 4.6 mm pressure. It requires a greater 
pressure than that of a single atmosphere to melt arsenic and 
arsenious oxide. All of the substances named above evapor- 
ate far below their fusion points. Hence, arsenious oxide ex- 
posed in a room, in wall paper, for example, may render the 
air of the room dangerous for inhalation. Ice evaporates at 
temperatures far below the fusion point. Housekeepers well 
know that clothes dry without thawing, and it is a familiar 
fact that snow-banks diminish in size when the weather is 
below zero. 

250. Heat of vaporization. — Heat that is consumed in the 
process of vaporization is called the heat of vaporization. 
The quantity of heat required to convert a gram of water at 
100° C. into steam without altering its temperature (which is 
the same as the quantity of heat generated by the condensa- 
tion of one gram of steam at 100°) is called the heat of 
vaporization of water. 

Experiment 5. — Let it be required to find the heat of vaporization of 
water. Find tlie mass in grams of the glass beaker or calorimeter C (Fig. 212), 
and since it will receive a small 
portion of the heat generated 
by the condensation of the 
steam, find its water equiva- 
lent by multiplying its mass 
by the specific heat of glass 
(.177). Represent this quan- 
tity by mi. Take in the 
calorimeter a certain known 
mass M of cold water at 
a known temperature t. 
When water in the flask A 
begins to boil, introduce the •^^^' ^-^^^ 

end of the delivery tube B into the water in C. The steam that passes 
through the tube is condensed on entering the cold water, and heats the 


water. When a considerable portion of the water in A has been 
vaporized, the temperature ti of the water in C is taken again, and the 
contents of the calorimeter are again weighed. The increase m in the 
mass of water in C is the mass of steam which has been condensed. 
Let L be the heat of vaporization. Then the whole quantity of heat 
generated by the condensation of m grams of steam is Lm and the quan- 
tity of heat imparted to the cold water in falling from 100° to ti° is 
m (100 — ti), or the total quantity of heat given to the calorimeter and its 
original contents is Lm + m (100 — ^i). The heat required to raise the 
calorimeter and its original contents from t to ^i is (ilf +mi) (^i ~t). 
But these two quantities are equal, hence 

Lm + m (100 - ^i) = (Jf + mi) {h - t); 
whence L = (J^ + »»,) (e, - Q - ^(100 - ^ 


In practice various precautionSj which need not here be 
detailed, are necessary. Careful experiments have determined 
the value of L for steam to be 536 (Kohlrausch) small 
calories ; that is, it requires 536 small calories of heat to 
convert one gram of water at 100° into steam at 100°, or 
536 calories per kilogram, and when the process is reversed 
536 calories per kilogram of steam are generated by the 

When water is converted into steam, the larger portion of 
the heat which disappears is consumed in separating the 
molecules so far that molecular attraction is no longer sen- 
sible ; a small portion — about Jg — is consumed in over- 
coming atmospheric pressure.^ The amount of work done in 
boiling is very great, as shown by the amount of heat 
consumed. Hence it requires a long time for the water to 

1 In this connection a brief discussion of some molecular hypotheses may prove of 
interest. The molecule is regarded as a collection of atoms. It may possess both 
translatory and rotary motion as a whole, and hence kinetic energy. It may possess 
potential energy, in virtue of the mutual attraction between it and other molecules 
and their relative distances, since the potential energy increases with both the force 
and the distance apart. This latter may be termed Mi^er-molecular energy. The 
atoms within the individual molecule may also possess all these kinds of motion and 
energy, and the total amount of this energy ivithin the molecule may be called intra- 
molecular energy. The work which is performed when either the inter- or intra- 



acquire the requisite amount of heat. This is a protection 
against sudden changes. 

Steam is a most convenient vehicle for the conveyance of 
heat of vaporization, i.e. jjotential energy, from the boiler to 
distant rooms requiring to be heated. For example, for every 
kilogram of steam condensed in the pipes of the radiator, 
536 calories, or heat enough to raise 5.36 kilograms (about 
12 lbs.) of ice-water to the boiling point, are generated. 

251. Distillation. 

Experiment 6. — Vessel A (Fig. 213), called a condenser^ contains a 
coil B, called a worm^ of copper tubing, terminating at one extremity at a. 
The other end of the 
tube, 6, projects through d \ 

the side of the vessel 
near its bottom. Near 
the top of the vessel pro- 
jects another tube, c, 
called the overflow^ with 
which is connected a 
rubber tube, e. This 
tube conveys the warm 
water which rises from 
the surface of the heated 
worm away to a sink or 
other convenient recep- 

Take a glass flask of 
a quart capacity, fill it 
three - fourths full of 
pond or bog water. Connect the flask by means of a glass delivery-tube 

Fig. 213. 

molecular energy is altered, or both are altered, is called disgregation tvork. The 
heat energy is regarded as the kinetic energy of the molecules in their vibratory 
motions, apart from any energy of rotation about their own centers of mass, and 
apart from inter- or intra-molecular energy. 

When heat is imparted to a body, it is distributed, in general, so as to produce, in 
varying relative quantities, four effects, viz. : (1) to raise the temperature by 
increasing the vibratory speed of the molecular motion ; (2) to perform external 
work ; (3) to perform inter-molecular work ; (4) to perform intra-molecular work. 
3 and 4 are classed as disgregation work. 


with the extremity a of the worm. Heat the water in the flask ; as soon 
as it begins to boil, commence siphoning cold water through a small tube, d^ 
from an elevated vessel, E, into the condenser. Inasmuch as the worm is 
constantly surrounded with cold water, the steam on passing through it 
becomes condensed into a liquid, and the liquid (called the distillate) 
trickles from the extremity h into a receiving vessel. The distillate is 
clear, but the water in the flask acquires a yellowish brown tinge as the 
boiling progresses, due to the concentration of impurities (largely of 
vegetable matter) which are held in suspension and solution in ordinary 
pond water. The apparatus used is called a stilly and the operation 

When a volatile liquid is to be separated from water, — for 
example, when alcohol is separated from the vinous mash after 
fermentation, — the mixed liquid is heated to its boiling- 
point, which is lower than that of water. Much more of the 
volatile liquid will be converted into vapor than of the water, 
because its boiling point is lower. Thus a partial separation 
is effected. By repeated distillations of the distillate, a 95 
per cent alcohol is obtained. 

Sbction X. 


A body becomes cold only by losing heat. As heat passes 
only from warmer to colder bodies, it is evident that the 
temperature of a body 'cannot fall below that of surrounding 
bodies, — for example, below the temperature of other bodies 
in the same room, — by the natural process of imparting heat 
to its neighbors. The temperature of a body, then, can be 
reduced below that of its neighbors only by some artificial 

The fact that heat must be consumed in the conversion 
of solids into liquids and liquids into vapors, because work is 
done, is turned to practical use in many ways for the 


purpose of producing artificial cold. The following experi- 
ments will illustrate this process. 

252. Heat consumed in dissolving. — Freezing mixtures. 

Experiment 1. — Prepare a mixture of 2 parts, by mass, of pulverized 
ammonium nitrate and 1 part of ammonium chloride. Take about 75 cc 
of water (not warmer than 8°C.), and into it pour a large quantity of the 
mixture, stirring it while dissolving with a test-tube containing a little 
cold water. The water in the test-tube will be quickly frozen. A finger 
placed in the solution will feel a painful sensation of cold, and a thermom- 
eter will indicate a temperature of about — 10° C. 

One of the most common freezing mixtures consists of 
3 parts of snow or broken ice and 1 part of common salt. 
The affinity of salt for water tends to produce liquefaction of 
the ice, and the resulting liquid dissolves the salt, hoth opera- 
tions consuming heat. 

253. Heat consumed in evaporation. — The heat consumed 
in vaporization is greater than that consumed in liquefaction ; 
for example, in the case of water it is greater in the ratio of 
536 : 80. Hence evaporation is the more efficient means 
of producing extremely low temperatures. Whatever tends 
to hasten evaporation tends to accelerate the reduction of 
temperature. Rapidity of evaporation increases with the tem- 
perature., extent of surf ace exposed, diminution of pressure, and 
dryness of the atmosphere (see page 292). The more volatile 
the liquid employed for evaporation, other things being equal, 
the more rapid the consumption of heat. 

Experiment 2. — Fill the palm of the hand with ether ; the ether 
quickly evaporates, and produces a sensation of cold. 

Experiment 3. — Place water at about 40° C. in a thin porous cup, 
such as is used in a Grove's battery, and the same amount of water at 
the same temperature in a glass beaker of as nearly as possible the same 
size as the porous cup. Introduce into each a thermometer. The com- 
paratively large amount of surface exposed by means of the porous vessel 


will so hasten the evaporation in this vessel, that, in the course of 10 to 15 
minutes, a very noticeable difference of temperature will be indicated by 
the thermometers in the two vessels. 

In warm climates water is frequently kept in porous 
earthen vessels in order that its temperature may be kept low 
enough by evaporation to render it suitable for drinking. 

Experiment 4- — Fill an atomizer, such as is used in the toilet for 
throwing a spray of cologne, with ether and throw a spray of the liquid 
continuously upon the bulb of a thermometer. In a very short time the 
temperature of the mercury will fall, in a warm room, to — 8° C. or lower. 

Water may be frozen by its own evaporation in the receiver of an 
air-pump from which the air (and consequently the air-pressure) is 
removed. A dish of sulphuric acid should be placed in the receiver 
to absorb the water-vapor. 

By evaporating liquid ethylene and liquid air under a pressure of 
4 mm, Olzewski produced a temperature of — 220°C.i The cold 
produced by the evaporation of liquid carbon dioxide in the air, 
when it is relieved from pressure, is sufficient to freeze the greater 
part of it, producing a solid mass like snow, which evaporates 
slowly, producing a temperature of — 90° C. 

254. Sjjheroidal state, 

A drop of water placed on a smooth metal surface heated above 
200° C. will not come in contact with or moisten the surface, but 
assumes the form of a fiat spheroid and rolls about like a ball or 
spins on its axis. It is said to be in the spheroidal state. When in 
this state the liquid does not boil ; indeed, its temperature is several 
degrees below its boiling point. The liquid globule rests upon a 
cushion of its own vapor and is buoyed up by it. When^ however, 
the heated metal cools and the vapor pressure is not great enough 
to sustain the globule, it comes in contact with the metal, its tem- 
perature rapidly rises to the boiling point, and it is quickly con- 
verted into steam. Showmen place red hot irons in their mouths 
and dip their moistened hands into melted lead or even melted iron, 

1 See Philosophical Magazine (Feb. 1895). Tlie announcement is just made in 
Nature that Olzewski has lately succeeded in liquefying hydrogen and producing a 
temperature of — 243° C. 


without injury. A layer of spheroidal fluid prevents contact of the 
flesh with the heated metal. 

Boutigny placed liquid sulphur dioxide, whose temperature when 
in the spheroidal state is below zero, in a red-hot platinum crucible ; 
it quickly assumed the spheroidal state, and drops of water let fall 
upon it quickly froze. Mercury can in like manner be frozen in a 
red-hot crucible by employing liquid nitrous oxide in the spheroidal 

Section XI. 


255. Dew-point. — Hygrometry treats of tlie state of the air 
with regard to the water vapor it contains. A given space, 
e.g. a cubic meter (it matters little whether there is air in the 
space or whether it is a vacuum), — can hold only a limited 
quantity of water vapor. This quantity depends on the tem- 
perature. The capacity of a space for water vapor increases 
rapidly with the temperature, being nearly doubled by a rise 
of 10° C. On the other hand if air containing a given quan- 
tity of water vapor be cooled, it will continually approach and 
finally reach saturation, since the lower the temperature, the 
less the capacity for water vapor. It is evident that air 
saturated with vapor cannot have its temperature lov^ered 
without some of the vapor being condensed into a liquid, 
which will appear, according to location and condition of 
objects within it, as dew, fog, or cloud} The temperature at 
which this condensation occurs is called the dew-point for air 
containing this proportion of water vapor. The dew point 
may be defined as the temperature of saturation for the 
quantity of water vapor actually present in the air. The 

-i Clouds formed at temperatures above 0° consist of minute spherical drops of 
water, ^^Lq to -^^-^ of an inch or more in diameter. Clouds formed at temperatures 
below 0° consist of minute ice spicules, which may increase in size and become 



greater the quantity of water vapor present in tlie air, 
the higher is its dew-point. Capacity for water vapor depends 
upon temperature ; dew-point depends upon quantity of vapor 

If the existing temperature be far above dew-point it in- 
dicates tliat the air can contain much more vapor than there 
is in it at the time, and the air is said to be dry. If the tem- 
perature of the air be little above 
dew-point, the air is said to be 
humid, which means that it 
can hold but little more vapor. 
The sensation of dryness experi- 
enced, especially in rooms heated 
artificially, does not depend 
upon the absolute quantity of 
water vapor present per cubic 

r"^'"" """T|"" "^L_|y foot. 
J The heat of a stove, for in- 

^ stance, dries the air of a room 

without destroying any of its 
water vajDor. In such a room, 
the lips, tongue, throat, and skin 
experience a disagreeable sensa- 
tion of dryness, owing to the 
rapid evaporation which takes 
place from their surfaces. This 
should be taken as nature's ad- 
monition to keep water in the stove urns, and in tanks con- 
nected with furnaces. 

The quantity of water vapor present in the air is expressed 
either (1) by the mass of vapor per unit of volume ; or (2) by 
the ratio between the quantity actually present and that which 
would be present if the air were saturated at the temperature 
of observation. The latter is the more common and more 

Fig. 214. 



useful method, aud this ratio is called the relative humidity, 
or simply " humidity '^ of the air. It is expressed in per- 
centages. Thus, relative humidity = 75 per cent., or 0.75, 
denotes that the air contains three-fourths the quantity of 
water vapor required to saturate it at the present tempera- 

256. Wet and dry hulh thermometer. 

The relative humidity of air is measured in various ways and by 
various devices. The instrument most commonly used is Mason's 
wet and dry bulb thermometer, or as it is frequently called, psy- 
chrometer. It consists (Fig. 214) of two thermometers mounted side 
by side a short distance apart, one having a dry bulb and the other 
a bulb covered with muslin, kept moist by capillary action through 

1 The student may be profited by a perusal of the following, copied from the daily 
meteorological report in tbe London Times. [Speed the time when our own govern- 
ment, in consideration of its educational value at least, shall issue daily a similarly 
complete meteorological bulletin.] 

The Times Office, 2 a.m. 

Keadings of Jordan Barometer (cor- 

Temperature and Htgrometric Condition of the 
Air in Londox. August 25, 26.- 

i August 25-26. 1891. • 
1 A.M. P.M. ^ 

To^hef^ ^ 

t 6 






I'. 2 















Weight of 



cubic feet 

of air. 

Power of 
Air (per 
10 cubic 

tion = 

2 a.m.. 









Per Cent. 


Minimum Temperature, 55 deg. Maximum Tempera- 
ture, 63 deg. 

Summit Station, (4,40/ ft. above sea level). 






At 320 




to 6. 


to 10. 

9 A.M. 
9 P.M. 








Maxinmm temperature, 43 5; minimum temperature, J 
Black bulb, 56. Sunshine, none. Rainfall, 0-905 in. 
Base Station (42 ft. above sea level). 






At 32° 







9 P.M. 








Maodmum temperature, 57-4; minimum temperature, 51-3. 
Black bulb, 110. Sunshine, 1 hour 19 min. Rainfall, 
0-520 in. 


conducting threads of lamp-wick from a vessel of water below. The 
dry bulb indicates the temperature of the air itself ; while the wet 
bulb, cooled by evaporation, shows usually a lower temperature 
according to the amount and rapidity of evaporation. The differ- 
ence in temperature of the two bulbs is greatest when the air is 
dry est. 

By experiment has been ascertained the relative humidity 
corresponding to 1, 2, etc. degrees difference between the two 
thermometers for any given temperature of the air. Empirical 
psychrometrical tables similar to the reduced table in the Appendix 
(p. 622) accompany this instrument. The observer reads the tem- 
perature of the air and ascertains the difference of the temperatures 
of the two bulbs, and from these two numbers determines by the 
table the relative humidity, and the dew-point of the air at that 
time and place. 

Not only does hygrometry play an important part in the science 
of meteorology, and consequently have important bearings upon 
many branches of industry, but it also has an intimate relation to 
the hygienic qualities of the atmosphere. The human body is much 
affected by the hygrometric state of the air. 

Section XII. 


There is always a tendency to equalization of temperature ; 
that is, heat has a tendency to pass from a warmer body to a 
colder, or from a warmer to a colder part of the same body, 
until there is an equality of temperature. 

There are commonly recognized three processes of diffusion 
of heat, — conduction, convection, and radiation. 

257. Conduction. 

Experiment 1. — Place one end of a wire about 10 inches long in a 
lamp-flame, and hold the other end in the hand. Heat gradually travels 
from the end in the flame toward the hand. Apply your flngers succes- 
sively at different points nearer and nearer the flame ; you find that the 
nearer you approach the flame, the hotter the wire is. 


The flow of heat through an unequally heated body, from 
places of higher to places of lower temperature, is called 
conduction; the body through which it travels is called a 
conductor. The molecules of the wire iii the flame have their 
motion quickened; they strike their neighbors and quicken 
their motion ; the latter in turn quicken the motion of 
the next ; and so on, until some of the motion is finally 
communicated to the hand, and creates in it the sensation of 

Experiment 2. — Fig. 215 represents a board on which are fastened, 
by means of staples, four wires : (1) iron, (2) copper, (3) brass, and (4) 
German silver. Place a lamp-flame where the wires 
meet. In about a minute run your fingers along 
the wires from the remote ends toward the flame, 
and see how near you can approach the flame on 
each without suffering from the heat. Make a list 
of these metals, arranging them in the order of their 

Experiment 3. — Go into a cold room, and place ^iq 215 

the bulb of a thermometer in contact with various 
substances in the room ; you will probably find that they have the same, 
or very nearly the same, temperature. Place your hand on the same 
substances ; they appear to have_ very different temperatures. This is 
due to the fact that some substances conduct heat away from the hand 
faster than others. Those substances that appear coldest are the best 
conductors. If you go into a room warmer than your body, all this is 
reversed ; those substances which feel warmest are the best conductors, 
because they conduct their own heat to your hand fastest. 

You learn that some substances conduct heat much more 
rapidly than others. The former are called good conductors, 
the latter poor conductors. Metals are the best conductors, 
though they differ widely among themselves. 

Experiment 4. — Nearly fill a test-tube with water, and hold it some- 
what inclined (Fig. 216), so that a flame may heat the part of the tube 
near the surface of the water. Do not allow the flame to touch the part 


of the tube that does not contain water. The water may be made to 
boil near its surface for several minutes before any change of the tem- 
perature at the bottom will be perceived. 

Liquids, as a class, are poorer con- 
ductors than solids. Gases are much 
poorer conductors than liquids. It is 
difficult to discover that pure, dry air 
possesses any conducting power. The 
poor conducting poAver of our clothing 
is due partly to the poor conducting 
power of the fibers of the cloth, but 
Fig. 216. chiefly to the air which is confined by it. 

Loose garments, and garments of loosely woven cloth, inas- 
much as they hold a large amount of confined air, furnish a 
good protection from heat and cold. Bodies are surrounded 
with bad conductors, to retain heat when their temperature is 
above that of surrounding objects, and to exclude it when 
their temperature is below that of surrounding objects. In 
the same manner double windows and doors protect from 

258. Convection in. gases. — Conduction takes place gradu- 
ally and slowly at best from particle to particle, the body 
and its particles being relatively at rest. Convection takes 
place when the body moves or there is relative motion be- 
tween its parts, thus carrying heat. 

Experiment 5. — Hold your hand a little way from a flame, beneath, 
on the side of, and above the flame. At which place is the heat most 
intense ? 

Experiment 6. — Cover a candle-flame with a glass chimney (Fig. 217), 
blocking the latter up a little way so that there may be a circulation of 
air beneath. Hold smoking touch-paper near the bottom of the chimney ; 
the smoke seems to be drawn with great rapidity into the chimney at the 
bottom ; in other words, the office of the chimney is to create what is 
called a draft of air. Notice whether the combustion takes place any 
more rapidfy with than without the chimney. 



Experiment 7. — Place a candle within a circle of holes cut in the cover 
of a vessel, and cover it with a chimney, A (Fig. 218). Over an orifice 
in the cover place another chimney, B. Hold a roll of smoking touch- 
paper over B. The smoke descends this chimney, and passes through 
the vessel and out at A. This illustrates the method often adopted to 
produce a ventilating draft through mines. Let the interior of the tin 
vessel represent a mine deep in the earth, and the chimneys two shafts 
sunk to opposite extremities of the mine. A fire kept burning at the 
bottom of one shaft will cause a current of air to sweep down the other 
shaft, and through the mine, and thus keep up a circulation of pure air 
through the mine. 

Fig. 217. 

Fig. 218. 

The cause of the ascending currents is evident. Air, on becoming 
heated, expands rapidly and becomes much rarer than the surrounding 
colder air ; hence it rises much like a cork in water, while cold air pours 
in laterally to take its place. In this manner winds are created. Sea 
and land breezes are convection currents. 

The so-called trade-ivinds originate in the torrid or heated 
zone of the earth. The air over the heated surface of the 
earth rises, and the colder air from the polar regions flows in 


on botli sideS; giving rise to a constant wind from the KE.^ 
in the northern hemisphere, and a wind from the S.E. in the 
southern hemisphere. Convection currents on the surface of 
the sun often attain a velocity of 100 miles per second. 

259. Change of temperature in vertical currents ascending 
from the earth. 

The lower air in contact with the heated surface of the earth 
acquires a certain temperature and a corresponding expansive force 
previous to its ascent. As it reaches higher altitudes, the pressure 
upon it becomes less ; it therefore expands, pushing away the sur- 
rounding air, until, as a result of its expansion, its expansive force 
is reduced to equality with the pressure upon it. It follows from 
the dynamical theory of heat, that in doing this work the ascending 
air must expend some of its energy ; i. e. the work is done by the 
expenditure of some of its heat ; hence, the ascending air is cooled 
by the very processes involved in its ascent. The rate of cooling 
thus produced is about 1° for 100 m of ascent. Such changes are 
called adiahatic; i.e. they are produced without a transfer of 
heat. * 

260. Ventilation. — Intimately connected with the topic 
convection, is the subject (of vital importance) ventilation, 
inasmuch as our chief means of securing the latter is through 
the agency of the former. The chief constituents of our 
atmosphere are nitrogen and oxygen, with varying quantities 
of water vapor, argon, carbon dioxide gas, ammonia gas, nitric 
acid vapor, and other gases. The atmosphere also contains in 
a state of suspension varying quantities of small particles of 
free carbon in the form of smoke, microscopic organisms, and 
dust of innumerable substances. All of these constituents 
except the first four are called imjmrities. Carbon dioxide 
is the impurity that is usually the most abundant and most 
easily detected ; so it has come to be taken as the measure 
of the purity of the atmosphere, though not itself the most 

1 The easterly component is due to tlie earth's rotation. 



deleterious constituent. Its chief harm arises from its diluent 
effect upon the life-giving oxygen. Pure out-door air contains 
about 4 parts of carbon dioxide by volume in 10,000. If the 
quantity rise to 10 parts, the air becomes unwholesome. 

Experiment S. — Ascertain by means of Wolpert's air-tester i (ap- 
proximately) the number of parts in 10,000 of carbonic acid gas (and 
thereby determine approximately the degree of pollution by 
respiration and combustion) in the air of the school-room. 

Clean the test-tube with water containing a little vinegar, 
and afterwards rinse thoroughly with clean water. Fill the 
clean test-tube with lime-water, even with the horizontal 
mark. Expel all the air possible from the rubber bulb A 
(Fig. 219), by pressing on it with the thumb ; then allow it 
to fill with air from the room. Insert the small glass tube 
B into the lime-water nearly to. the bottom. Expel the 
air in the bulb again with moderate rapidity, so that the 
bubbles may rise nearly to the top of the tube C ; but do 
not allow the liquid to overflow. Continue the pressure 
until you have withdrawn the tube from the liquid, when 
you will allow the bulb to refill with air of the room. At 
the end of each expulsion place the bottom of the test- 
tube .on a sheet of white paper in good daylight, and look 
vertically down through the liquid at the black mark on the 
bottom of the test-tube ; repeat the process, being careful 
not to affect the result more than is necessary with your 
breath, until the turbidity of the lime-water renders the 
mark invisible. " If the mark become obscured after filling „ ^^ 
the bulb ten or fifteen times only, the air of the apartment 
is unfit for continuous respiration. With good air the bulb must be filled 
twenty-five times and upwards. The normal amount in pure out-door 
air is 3 to 5 parts per 10,000." 

Ascertain, by comparing your results with those given by Prof. Wol- 
pert in the table below, the number of parts of carbon dioxide in 10,000 
of the air of your room. Kepeat the experiment, taking air at different 
elevations in the room. 


1 This instrument is vised almost exchisively by inspectors of scliool and other 
public buildings in the State of Massachusetts. 




















per 10,000. 


per 10,000. 


per 10,000. 


per 10,000. 

























































































































Carbon dioxide is about one and one-half times heavier than 
air at the same temperature ; consequently, when both have 
the same temperature, and the former is very abundant, it 
tends to sink beneath the air, in which large quantities of 
this gas are generated. 

The knowledge of this fact has led many to suppose that a 
means for the escape of impure air need be provided only 
near the floor of a room. But it should be remembered that 
(1) the tendency of carbon dioxide, unless present in excessive 
quantities, is to diffuse itself equally through a body of air ; 
but (2) when it is heated to a temperature above that of the 
surrounding air, as when generated by flames, or when it 
escapes in the warm breath of animals, it is lighter than the 
air, and consequently rises. If this impure air could escape 
at the ceiling while fresh air entered at the floor, the ventila- 
tion would be good. But usually this fresh air must be 
warmed; and in passing over a stove, furnace, or steam 
radiator, its temperature will generally become higher than 



that of the impure air, so that it Avill rise above the latter, 
and pass out at a ventilator in the ceiling, leaving the floor 
cold ; hence, in high school-rooms the most impure air is 
often found half-way up. 

The quantity of fresh air introduced must be great enough 
to dilute the impurities till they are harmless. An adult 
makes about 18 respirations per minute, expelling from his 
lungs at each expiration about 500 cc of air, over 4 per cent 
of which is carbon dioxide. At this rate, about 9,000 cc of air 
per minute become unfit for respiration ; and to dilute this 
sufiiciently, good authorities say that about 100 times as much 
fresh air is needed ; or, for 
proper ventilation, about a 
cnhic meter of fresh air per 
minute is needed for each per- 
son^ or, in British measures, 
2,000 cubic feet per hour. 








Pure outdoor air. 

Fig. 220 represents a scheme 
for heating a room by steam, 
and ventilating it by con- 
vection. Steam is conveyed 
by a pipe from the boiler 
to a radiator box just be- 
neath the floor of the room. 
The air in the box becomes 
heated by contact with and 
radiation from the coil of 
pipe in the box, and rises 
through a passage opening 
into the room by means of a 
register near the floor at C, a 
supply of pure air being kept 
up by means of a tubular 
passage opening into the box 
from the outside of the building. Thus the room is furnished with 
"pure warm air, which, mmgiing with the impurities arising from the 
respiration of its occupants, serves to dilute them and render them 

Fig. 520. 


less injurious. At the same time, the warm and partially vitiated 
air of the room passes through the open ventilator A into the 
ventilating-flue, and escapes, so that in a moderate length of time 
a nearly complete change of air is effected. It is evident that on 
the coldest days of winter the convection is most rapid ; indeed, it 
may be so rapid that the air cannot be heated sufficiently to render 
the room near the floor comfortable. At such times the ventilator 
A may be closed, while the ventilator B is always open. The 
heated air rises to the top of the room and, not being able to escape, 
crowds the colder air beneath out at the ventilator B. No system 
of ventilation dependent wholly on convection is adequate properly 
to ventilate crowded halls ; air is too viscous and sluggish in its 
movements. In such cases ventilation should be assisted by some 
mechanical means, such as a blower or fan, worked by steam or 
water power. 

261. Convection in liquids. 

Experiment 9. — Fill a small (6 ounce) thin glass flask with boiling hot 
water, color it with a teaspoonful of ink, stopper the flask, and lower it 
deep into a tub, pail, or other large vessel filled with cold water. With- 
draw the stopper, and the hot, rarer, colored water will rise from the 
flask, and the cold water will descend into the flask. The two currents 
passing into and out of the neck of the flask are easily distinguished. 
The colored liquid marks distinctly the path of the heated convection 
currents through the clear liquid and makes clear the method by wdiich 
heat, when applied at the bottom of a body of liquid, becomes rapidly 
diffused through the entire mass notwithstanding that liquids are poor 

Experiment 10. — Ffll again the flask with hot colored water, stopper, 
invert, and introduce the mouth of the flask just beneath the surface of a 
fresh pail of cold water. Withdraw the stopper with as little agitation 
of the water as possible. What happens ? Explain. 

Ocean currents, e.g. the gulf stream, are convection currents. 
Liquids are also cooled by convection currents. When the 
air above the surface of a pond, for instance, is cooler than 
the surface v^^ater, the latter gives heat to the former, cools, 
becomes denser, and sinks. Meanwhile the warmer and rarer 
water below rises, and in this way the eutire body is kept at 


an approximately uniform temperature until it reaches 4° C, 
at which point convection ceases. 

262. Radiation. — In radiatio7i a hotter body loses heat, 
and a colder body is warmed, through the transmission of 
undulatory motion in a medium called the ether, ivhich is not 
itself heated thereby. It is neither a mass nor a molecular 
transference of heat; in fact heat itself is not transferred 
by radiation at all. Heat generates radiation (ether wave 
motions) at one place, and the body which obstructs these 
waves transforms the energy of their motion, or as it is com- 
monly called radiant energy, into heat. In this manner the 
earth is heated by the sun, though no heat passes between 
them. In this manner radiant energy passes through glass 
and slabs of ice without heating them much, since they offer 
little obstruction to the passage of ether waves. All bodies 
emit radiant energy, and there is an exchange of energy 
between bodies by radiation, going on at all times. This 
mode of transmission of energy is the most important of all, 
and will be treated fully in the next chapter. 

Section XIII. 


263. Thermo-dynamics defined. — TherTno-dynamics treats 
of the relation between heat and mechanical ivork. One of the 
most important discoveries in science is that of the equivalence 
of heat and ivork; that is, that a definite quantity of mechanical 
work, ivhen transformed ivithout luaste, yields a definite quantity 
of heat ; and conversely, this heat, if there be no luaste, can 'per- 
form the original quantity of mechanical work. 

264. Transformation, correlation, and conservation, of energy. 
— The proof of the facts just stated was one of the most 
important steps in the establishment of the grand twin con- 


ceptions of modern science : (1) that all kinds of energy are 
so related to one another that energy of any hind can he trans- 
formed into energy of any other kind, — known as the doctrine 
of CORRELATION OF ENERGY ; (2) that when one form of 
energy disappears, its exact equivalent in another form, ahuays 
takes its place, so that the sum total of energy is unchanged, — 
known as the doctrine of conservation of energy. 

These two doctrines are admirably summarized by Maxwell 
as follows : "^The total energy of any body or system of bodies is 
a quantity which can neither be increased nor di7iiinished by 
any mutual action of these bodies, though it may be transforvied 
into any of the forms of which energy is susceptible.'''' Since all 
bodies of matter in the universe constitute a system, it 
follows from the above that the sum total of energy in the 
universe is a constant quantity. Neither creation nor annihi- 
lation of energy is possible through any agency known to 
man. These doctrines constitute the corner stones of modern 
physical science. Chemistry teaches that there is a conser- 
vation of matter, i.e. that matter is neither creatable nor 
annihilable through any known natural agency or process. 

265. Jolliers experiment. — Two laws of ther mo-dynamics. — 
The experiment to ascertain the " mechanical value of heat," 
as performed by Dr. Joule of England, was conducted about 
as follows : 

A copper vessel, B (Fig. 221), was provided with a paddle 
wheel (indicated by the dotted lines), which rotated about a 
vertical axle, A. The axle was rotated by the weights E and 
F, the cord of each being so arranged that each weight, in 
falling, rotated the axle in the same direction. By turning 
the crank above A the weights are raised to any desired hight 
measured on the scales G and H. 

The resistance offered by the water to the motion of the 
paddles was the means by which the mechanical energy of 
the weights was converted into heat, which raised the tem- 



perature of the water. Taking two bodies whose combined 
mass was, e.g.^ 80 K, he raised them a measured distance, e.g. 
53m high; by so doing 4240 kgm of work were performed upon 
them, and consequently an equivalent amount of energy was 
stored up in them, ready to be converted, first into that of 
mechanical motion, then into heat. He took a definite mass 
of water to be agitated, e.g. 2 K, at a temperature of 0° C. 
After the descent of the weights, the water was found to have 
a temperature of 5° C. ; consequently the 2 K of water must 
have received 10 calories of heat (careful allowance being 

Fig. 221. 

made for all losses of heat), which is the number of calories 
that is equivalent to 4240 kgm of mechanical energy ; or one 
calorie is equivalent to 4^4 %^^ (commonly taken as 4.2 X 10'' 
ergs per gram degree) of mechanical energy. 

In other words, to produce the quantity of heat required to 
raise 1 kilogram of water through 1° C, 4^4- hilogr ammeters of 
mechanical energy must he consumed. What the experiment 
really shows is that whenever a certain quantity of mechanical 
energy is converted into heat, the number of thermal units 
produced is always proportional to the mechanical energy 
consumed, or to the work done. This is embodied in the 


first law of thermo-dynamics^ which is expressed as follows ■. 
" When equal quantities of ^mechanical effect are produced by 
a?ii/ means whatever from 2)ureli/ thermal sources, or lost in 
purely thermal effects, equal quantities of heat are put out of 
existence, or are generated.'''' It is apparent that heat, being a 
form of energy, may be measured in ergs. In this way the 
erg is regarded as the mechanical nnit of heat. The advan- 
tage of this is found in the fact that we are often in the 
position of having to solve problems in which heat and work 
enter as terms to be added together. The existence of 
quantitative correlations between all the various forms of 
energy imposes upon men of science the duty of bringing all 
kinds of physical quantities to one common scale of com- 
parison, as is attempted in the absolute system. 

Mechanical energy can be wholly converted into heat, but 
it can be demonstrated that heat under the jnost favorable 
circumstances conceivable, even with the use of an ideally 
perfect heat engine {i.e. one which wastes no heat), can never 
be wholly converted into work. A portion — a large portion 
— of the heat employed must be given up to some substance 
termed technically a '' refrigerator,'' which in some form is a 
necessary adjunct to every heat-engine, and that portion still 
exists as heat. This is a practical deduction from the so- 
called Second Law of Thermodynamics ; ^ viz., " It is impos- 
sible to derive m^echanical effect from any portion of m^atter by 
cooling it below the temperature of the coolest surrounding ob- 
jects. ^^ The ratio of the heat converted into work and the 
entire heat employed is called the efficiency of the engine. 
" For any boiler-pressure " (of a steam-engine) " which it is 
safe to employ in practice, it is not possible, even with a 
perfect engine, to convert into work more than about fifteen 
percent, of the heat used." ^ — Anthony and Bkackett. 

1 Any adequate discussion of this law would take us beyond the limits proposed 
for this book. This subject is more fully treated in the works of Barker and Daniell. 



266, Meclianical equivalent of heat. — As a converse of the 
above it may be demonstrated by actual experiment that the 
quantity of heat required to raise 1 K of water from 0° to 
1° C. will, if converted into work, raise a 424 K weight 1 m 
high, or 1 K weight 424 m high. According to the British 
system, the same fact is stated as follows : The quantity of 
heat that will raise the temperature of 1 pound of water from 
60° to 61° F. will raise 772.55 pounds 1 foot high. The 
quantity, 424 kgm, is called the mechanical equivalent of one 
calorie, or Joule'' s equivalent (abbreviated simply J). J is 
the number of units of energy or work per unit of heat. Or we 
may say that one calorie is the thermal equivalent of 424 kgm 
of work,^ or the thermal equivalent of 1 kgm is ^^^ calorie. 


Equivalent in 


Equivalent in 

































If we denote by H the number of calories, and by W the 
number of kilogrammeters of mechanical energy, then the 


ratio — =r - (a constant) = ^^^ ; whence H = — . 

1 A knowledge of the exact numerical value is of great scientific and practical 
importance. The results as obtained by Eowland (1879) with improved apparatus 
and by improved methods, though the same in principle as that employed by Joule, 
are doubtless more accurate and are likely to come into general use for engineering 
and scientific purposes. 

" The following table gives the number of kilogrammeters required to raise one 
kilogram of pure water from f to t° + 1, as found by Rowland, for the latitude of 
Baltimore, and at sea-level. At Baltimore, ^ = 980.05 cm. To reduce to any other 
latitude chan Baltimore, add, for lat. 30°, 0.34 kgm ; lat. 40°, 0.08 kgm ; lat. 50°, 
— 0.41 kgm. The value of J ordinarily used in engineering computations is 424 kgm. 
As most measurements with which this value of J is employed are made at about 15° 
to 25° C, this value is too small by one-half per cent, or more." 


Section XIY. 

thermodynamics continued. steam-engine. 

267. Description of a stPMin-engiyie. — A steam-engine is a 
macliine in which the elastic force of steam is the motive 
agent. Inasmuch as the elastic force of steam is entirely due 
to heat, the steam-engine is properly a heat engine ; that is, it 
is a machine by means of which heat is continuously trans- 
formed into work, or the energy of mass motion. 

The modern steam-engine consists essentially of an arrange- 
ment by which steam from a boiler is conducted to each side 
of a piston alternately; and then, having done its work in 
driving the piston to and fro, is discharged from each side 
alternately, either into the air or into a condenser. The 
diagram in Fig. 222 will serve to illustrate the general fea- 
tures arid the operation of a steam-engine. The details of the 
various mechanical contrivances are purposely omitted, so as 
to present the engine as nearly as possible in its simplicity. 

In the diagram, B represents the boiler, F the furnace, S 
the steam-pipe through which steam passes from the boiler to 
a small chamber VC, called the valve-chest. In this chamber 
is a slide-valve V, which, as it is moved to and fro, opens and 
closes alternately the passages M and N leading from the 
valve-chest to the cylinder C, and thus admits the steam 
alternately each side of the piston P. When one of these 
passages is open, the other is always closed. Though the 
passage between the valve-chest and the space in the cylinder 
on one side of the piston is closed, thereby preventing the 
entrance of steam into this space, the passage leading from 
the same space is open through the interior of the valve, so 
that steam can escape from this space through the exhaust- 
pipe E. Thus, in the position of the valve represented in the 
diagram, the passage N is open, and steam entering the cylin- 



der at the top drives the piston in the direction indicated by 
the arrow. At the same time the steam on the other side of 
the piston escapes through the passage M and the exhaust- 
pipe E. While the piston moves to the left, the valve moves 
to the right, and eventually closes the passage N leading 
from the valve-chest and opens the passage M into the same, 
and thus the order of things is reversed. 


Fig. 222. 

Motion is communicated by the piston through the piston- 
rod E, to the crank G, and by this means the shaft A is 
rotated. Connected with the shaft by means of the crank H 
is a rod W which connects with the valve V, so that, as the 
shaft rotates, the valve for the greater part of its stroke is 
made to slide to and fro, in a direction opposite to that of 
the motion of the piston. 


The shaft carries a fly-wheel W. This is a large, heavy 
wheel, having the larger portion of its mass located near 
its circumference ; it serves as a reservoir of energy, which 
is needed to make the rotation of the shaft and all other 
machinery connected with it uniform, so that sudden changes 
of velocity resulting from sudden changes of the driving 
power or resistances may be avoided. By means of a belt 
passing over the wheel W motion may be communicated 
from the shaft to any machinery desirable. 

268. Condensing and non-condensmg-engines} — Sometimes 
steam, after it has done its work in the cylinder, is conducted 
through the exhaust-pipe to a chamber Q, called a condenser, 
where, by means of a spray of cold water introduced through 
a pipe T, it is suddenly condensed. This water must be 
pumped out of the condenser by a special pump, called tech- 
nically the air-iDumjp ; thus a partial vacuum is maintained. 
Such an engine is called a condensing-engine. Its advantage 
is obvious, for if the exhaust-pipe, instead of opening into a 
condenser, communicate with the outside air, as in the non- 
condensing engine^ the steam is obliged to move the piston 
constantly against a resistance arising from atmospheric 
pressure of 15 pounds for every square inch of the surface of 
the piston. But in the condensing engine a large portion of 
the pressure on the exhaust side of the piston is removed and 
an equivalent portion of the pressure on the steam side is 
utilized and made to do useful work. Tn well proportioned 
condensing apparatus the pressure on the exhaust side may 
be reduced 90 per cent., so that the moving piston instead of 
working against a resistance of 15 lbs. meets with a resistance 
of only 1.5 lbs. per square inch. 

269. Steam gauge. ■ — An instrument called a steam gauge 
is connected with the boiler. It ^measures the excess of the 

1 The terms, low-pressure and high-pressure engines, are not distinctive as applied 
to engines of tlie present day. 



pressure of the steam at any instant above the atmospheric 
pressure. The absolute pressure of the steam (i.e. measured 
from zero) is the pressure indicated by the steam gauge ^^^i^s 
the pressure of the atmosphere at the time. 

270. Compound condensing or douhle-cylinder engine. — This 
engine has two cylinders, each like that of a simple engine. 
One, A (Fig. 223), called the high-pressure cylinder^ receives 
steam of very high pressure directly from the boiler through 


Fig. 223. 

the orifice V. The steam, after it has done work in this 
cylinder, passes through the steam-port E into cylinder B, 
called the loiu-pressure cylinder. Cylinder B is larger than 
cylinder A. The steam which enters cylinder B possesses 
considerable pressure, and is therefore capable of doing con- 
siderable work under suitable conditions. It should be borne 
in mind that in order that steam may do work in any cylinder, 
it is necessary that an inequality in the pressure of the steam 


each, side of the piston should be maintained ; just as an 
inequality of level, i.e. a head, is essential to water-power. 
The steam, after it has done its work in cylinder B, passes 
through a port C into a condenser (not represented in the 
figure), where it is suddenly condensed or let down to a very 
low pressure. If a vertical glass tube were led from the con- 
denser to a vessel of mercury below, the mercury would 
ordinarily stand about 25 inches high in the tube, which 
would show that the pressure of the steam against which the 
steam when it enters cylinder B does work, is only about 
one-sixth of an atmosphere. Much energy is economized by 
the compound engine. 

271. The locomotive. — The distinctive feature of the loco- 
motive engine is its great steam-generating capacity relatively 
to its size and weight, which are necessarily limited. To do 
the work ordinarily required of it, from three to six tons of 
water must be converted into steam per hour. This is 
accomplished in two ways : first, by a rapid combustion of 
fuel (from a quarter of a'ton to a ton of coal per hour); 
second, by bringing the water in contact with a large extent 
(about 800 square feet) of heated surface. The fire in the 
" fire-box " A (Plate II) is made to burn briskly by means of a 
powerful draft which is created in the following manner : 
The exhaust steam, after it has done its work in the cylinders 
B, is conducted by the exhaust-pipe C to the smoke-box D, 
just beneath the smoke-stack E. The steam, as it escapes 
from the blast-pipe F, pushes the air above it, and drags by 
friction the air around it, and thus produces a partial vacuum 
in the smoke-box. The external pressure of the atmosphere 
then forces the air through the furnace grate and hot-air 
tubes G-, and thus causes a constant draft. The large extent 
of heated surface is secured as follows : The water of the 
boiler is brought not only in contact with the heated surface 
of the fire-box, but it surrounds the pipes G (a boiler usually 


contains about 150). These pipes are kept hot by the heated 
gases and smoke, all of which must pass through them to the 
smoke-box and smoke-stack. 

272. Fower of a steam engine. — The horse-power of a 
steam engine is calculated by means of the following formula, 

(Mean effective pressure in lbs. per sq. in. on piston X area 
of piston in sq. in. X length of stroke in ft. X number of strokes 
per min.) -r- 33,000. 

The steam engine, with all its merits and with all the 
improvements which modern mechanical art has devised, is 
an exceedingly wasteful machine. The best engine that has 
been constructed utilizes less than 15 per cent, of the heat 
energy generated by the combustion of the fuel. 


1. What kind of engine {i.e. condensing or non-condensing) is that 
which produces loud puffs ? What is the cause of the puffs ? 

2. Why does the temperature of steam suddenly fall as it moves the 
piston ? 

3. What do you understand by a ten horse-power steam-engine ? 

4. Upon what does the power of a steam-engine depend ? 

5. Is the compound engine a condensing or a non-condensing engine ? 
Which is the locomotive engine ? 

6. The area of a piston is 500 square inches, and the average unbal- 
anced steam pressure is 30 pounds per square inch ; what is the total 
effective pressure ? Suppose that the piston travels 30 inches at each 
stroke, and makes 100 strokes per minute ; 40 per cent being allowed for 
wasted energy, what power does the engine furnish, estimated in horse- 
powers ? 

7. A leaden bullet of mass 56 g strikes a target with a velocity of 300 
meters per second ; its temperature is 16° C. If two-thirds of the energy 
of the bullet be used in raising its temperature, determine its final tem- 

8. Can ice at 0° C, and under ordinary atmospheric pressure, have its 
temperature raised ? Explain. 


9." Eind the resulting temperature (C.) of the following mixtures: — 
a. 5 K of snow at 0° with 25 K of water at 28°. 
6. 4 K of ice at — 10° with 30 K of water at 50°. 
c. 10 K of iron at 200° with 2 K of ice at 0°. 

10. How many thermal units are required to change 5K of ice at 
— 10° C. into water at 10° ? 

11. If 30 g of steam at 100° C. be passed into 400 g of ice-water at 
0° C, what will be the temperature of the mixture ? 

12. A building is heated by hot-water pipes. How does heat get from 
the furnace of the boiler to a person in the building ? 

13. A building is heated by steam pipes. How does heat get from the 
furnace to objects in the building ? 

14. A rod of copper at 0° C. measures 10 ft, ; its length at 100° C. is 
0.191 inch greater. Find the coefficient of expansion of copper. 

15. A silver rod at 0° C. is 10 ft. long ; find its length at 100° C. 

16. A cubic meter of air at 100° C. is cooled down to 0°, and at the 
same time its pressure is halved ; determine its new volume. 

17. A copper ball weighing 3K, taken out of a furnace and plunged 
into 8 k of water at 10° C, heated the water to 25° ; find the temperature 
of the furnace. 

18. If the heat yielded by 1 K of water in cooling down from 100° to 
0° C. were employed in heating 10 K of mercury, initially at 20°, to what 
temperature would the mercury be raised ? 

19. A kilogram of ice at 0°C. is thrown into 6.3 K of water at 15°; 
when the ice is melted, the temperature of the water is 2°. Eind the 
heat of fusion of ice. 

20. A mass of 93.3 g of copper at 80° C. is immersed in 560 g. of water 
at 10°, and raises the temperature of the water to 20° ; find the specific 
heat of copper. 




Section I. 


Owing to the peculiarity of the subject to be treated in 
this, the third and final natural division of Physics, it is 
deemed expedient to state at the outset some leading proposi- 
tions, whose truth must be assumed as the basis for the study 
of a large group of natural phenomena. The demonstrations 
of the validity of these several assumptions must, however, 
be deferred to their proper place in connection with the study 
of the phenomena themselves. 

273. The ether. — We know matter by its properties as 
perceived by means of our senses ; in other words the exist- 
ence of any form of matter is to us only an inference from 
the phenomena to which it gives rise. By evidence of pre- 
cisely similar nature are we led to believe in the existence of 
a medium called the ether, pervading all space and penetrating 
between the molecules of matter, which are imbedded in it 
and surrounded by it as the earth is surrounded by its atmos- 
phere. We cannot see, hear, feel, taste, smell, exhaust, weigh, 
or measure it, and yet all this, paradoxical as it may seem, 


furnislies absolutely no evidence that it does not exist. Briefly 
stated, the proof of its existence is this : it furnishes the 
basis for the sole conceivable explanation of very many physical 

Phenomena occur just as they would occur if all space were 
filled with an intangible and invisible medium capable of 
transmitting motion and energy, and we can account for all 
these phenomena on no other hypothesis ; hence our belief in 
the existence of the medium. The evidence of the existence 
of ether is as strong and direct as that of the existence of 
air. The eye is an ether sense-organ just as the ear may be 
called an air sense-organ, or the hand a sense-organ for the 
appreciation of grosser forms of matter.^ The ether is a 
Tnedium for the transmission of energy in the form of vibrations. 

In its structure the ether is assumed to be excessively fine- 
grained, ^'Differing from water, glass and metals in being 
very much more finely grained in its structure " ^ (Lord Kelvin). 
It possesses rigidity,^ and in this respect is like a solid. Bodies 
of matter, even so large as the planets, pass freely through it, 
encountering little resistance ; therein it is like a perfect 
fluid. It is almost perfectly elastic and incompressible. 

274. Radiation. Radiant energy. — The transmission of 
energy by means of periodic disturbances in the ether is called 
radiation; energy so transmitted is called radiant energy; 

1 " Instead of beginning by saying tbat we know nothing about the ether, I say 
that we know more about it than we do about air or water, glass or iron, — it is far 
simpler ; there is far less to know. Its natural history is far simpler than that of 
any other body." — Lord Kelyin, in lectures on Molecular Dynamics at Johns 
Hopkins University (1884). 

2 " The ether is practically a homogeneous solid, — in other words an exceedingly 
fine-grained solid, so finely grained that it is practically homogeneous for portions 
exceedingly small in linear dimensions in comparison with the wave-length. But no 
degree of smallness will dispense with the to and fro motion of the elastic solid rela- 
tively to the imbedded molecules." 

3 Calculation leads us to infer that its rigidity is about lO-^ that of steel, and its 
density 936 x IQ-^i that of water at 4° C, or equal to that of our atmosphere at a hight 
of 210 miles, — a density vastly greater than that of the same atmosphere ' in the 
interstellar spaces. — Maxwell. 


and the body emitting energy in tliis manner is called a 
radiator. The precise nature of the periodic disturbances, 
whether they be due to changes of position in the ether, or to 
alternation between opposite conditions (e.g. such as succes- 
sive local states of strain or distortion and release therefrom) 
is unknown to us. We do know, however, that the laws 
according to which these changes take place are those of 
wave-motion. Space is traversed at all times and in all direc- 
tions by myriads of ether-waves of all possible lengths. The 
all-pervading ether can be set in vibration by the motion of the 
molecules of ordinary matter. This local disturbance creates 
ether-waves, and by these waves energy is transferred from 
place to place by the process, as stated above, called radia- 
tion. Eadiant energy can be transformed into any other 
form of energy, and therefore offers no exception to the doc- 
trine of correlation of energy. 

Just how vibrations of particles of matter create ether- 
waves, and what constitutes a wave of ether, are things of 
which our knowledge is as yet very deficient. It must be 
remembered that ether is a substance very unlike ordinary 
matter, and, therefore, reasoning by analogy must often fail. 
It is generally supposed that ether-waves are not waves of 
compression and rarefaction, like those of sound-waves in air. 

Furthermore, the vibrations which occur in the ether are 
not longitudinal like those of the air particles during the 
passage of sound-waves, but are transversal and somewhat 
analogous to the motions of particles of water in water-waves. 
That is, the vibrations in ether are at right angles to the 
direction in which the wave is propagated, and are therefore 
parallel to the wave-front. 

275. Effects of radiant energy. — When radiant energy is 
received upon the surfaces of our bodies, warmth is felt ; when 
upoUjthe bulb of a thermometer, rise of temperature is indi- 
ca;ted ; when by the eye, the sense of sight may be affected ; 


if, upon sensitive photographic plates, upon the leaves of 
plants, and upon various chemical mixtures, chemical changes 
may be promoted. Thus it seems that when ether-waves 
impinge upon objects .their energy is transformed, producing 
effects of different kinds, which are determined by the nature 
of the body upon which they fall. The effect which most 
concerns us is that produced when the radiations strike the 
eye and become the means, through this organ, of awakening 
in the brain the sensation which we call Light. 

Section II. 


276. Light defined. Hypotheses. — Physiologically speak- 
ing, light is the sensation of sight. Physically considered, it 
is that agent which, by its action on the retina of the eye, excites 
in us the sensation of vision. Two leading hypotheses ^ regard- 
ing the nature of light have been propounded, which are 
totally different in character. One is the so-called eynission or 
corpuscular hypothesis which was supported by Descartes 
(1629), Newton (1672), and most physicists up to the early 
part of the present century. It assumes that a luminous 
body {e.g. the sun) emits minute material particles (cor- 
puscles) which travel through space in all directions with 
immense velocity ; these particles by their impact upon the 
nerve-woven retina produce the sensation of sight. As a 
rose emits minute particles which, reaching the nostrils, 
enable us to smell the rose, so a star is supposed to emit par- 

■ 1 The Platonists maintained that the sensation of light was produced and vision 
effected by something which was emitted from the eye to tlie object, and the sense of 
vision was explained by the analogy of touch. " The light from the sun, the twink- 
ling of the stars, the colors of the rainbow, and the various hues of the floor of 
nature remain the same as when they gladdened the heart of Noah ; but how have 
the explanations of the phenomena varied ! " 


tides of light which, on reaching the eye, enable us to see 
the star. 

This hypothesis is now discarded by scientists ; the reasons 
for its abandonment will appear further on. The theory 
which obtains at the present time, called the undulatory or 
wave-theory,^ is based upon the hypothesis that energy is 
transmitted from body to body, e.g. from the sun to the earth 
(and the reverse), in the form of vibrations or wave-action in 
the all-pervading ether. In this connection it should be 
borne in mind that the evidence of the correctness of any 
theory is its exclusive competence to explain and coordinate 
phenomena. It is not claimed that all phenomena have been 
fully explained by the wave theory; it is merely claimed 
that all we know at the present time about light is in perfect 
accord with it. It will be observed that both theories recog- 
nize the fact that light is essentially dynamic. According to 
the latter theory, light is that vibration of the ether which may 
be appreciated by the organ of sight. '^ 

277. Luminous and illuminated objects. — Some bodies are 
seen by means of light-waves which they generate and emit ; 
e.g. the sun, a candle flame, and a '' live coal " ; they are 
called luminous bodies. Other bodies are seen only by means 
of light-waves which they receive from luminous ones and 
reflect to the eye, and, when thus rendered visible, are said to 
be illumhiated ; e.g. the moon, a man, a cloud, and a "dead'' 

1 The first person who presented the wave-theory of light in a definite shape was 
Huygens, in a work published in 1690 under the title of Traite de la Lumiere. The 
theory was thoroughly established by Young and Fresnel between the years 1800 and 

2 It will be shown further on, that not all ether-waves are capable of afEecting 
the sight, hence for the purpose of distinction we apply the term light-waves to those 
ether-waves only which are capable of producing vision. It is strongly recommended 
that the student in beginning this branch of science make use of the term Hght-icaves 
instead of light except when such usage would lead to an inconvenient circumlocu- 
tion, in order that he may have strongly impressed upon his mind the fact that when 
he is dealing with light he is dealing with waves. 



278. Light itself invisible. — Light makes visible to us 
luminous or illuminated objects, light-waves from which 
actually reach our eyes ; but if we look across the line of 
direction of a series of light-waves, termed the path of the 
light, we cannot see the light. If we appear to see a sun- 
beam admitted through a key-hole or knot-hole, and travers- 
ing a darkened room, it is only because it is made to reveal its 
track by illuminating the dust motes floating in the air. If 
the air in a certain space be cleansed of dust, the path of a 
sunbeam through the space will be totally imperceptible.^ 

279. Light-waves travel in straight 
lines. — The path of light-waves ad- 
mitted into a darkened room through 
a small aperture, as indicated by the 
illuminated dust, is perfectly straight. 
An object is seen by means of light- 
waves which it sends to the eye. A small 
object placed in a straight line between 
the eye and a luminous point may in- 
tercept the light-waves in that path, 
and the point become invisible. Hence 
we cannot see around a corner, or 
through a bent tube. 

280. Ray, beam, pencil. — Any line 
EE (Fig. 224) which pierces the surface 
of an ether-wave ab perpendicularly, is 
called a ray. The term '' ray " is but 

an expression for the direction in which motion is propagated, 
and along which the successive effects of ether-waves occur. '^ If 
the wave-surface a^V be a plane, the rays R'E' are parallel, 

1 See Tyndall's Fragments of Science, p. 277. 

2 In dealing with certain phenomena {e.g. reflection of light) we may, to facilitate 
our study, consider the light as propagated, in straight lines or rays ; but we must 
hear in mind that a ray has no material or physical existence, for it is a wave that 
is propagated, not a ray. 


and a collection of such rays is called a beam. If the wave- 
surface <:t"6" be spherical, the rays E."R" have a common point 
at the center of curvature ; and a collection of such rays is 
called a 'pencil. 

281. Transparent, translucent, and opaque substances. — Sub- 
stances are transparent, translucent, or opaque, according to 
the manner in which they act upon the light-waves which 
are incident upon them. Generally speaking, those sub- 
stances are transparent that allow other objects to be seen 
through them distinctly, e.g. air, glass, and water. Those 
substances are translucent that allow light-waves to pass, but 
in such a scattered condition that objects are not seen dis- 
tinctly through them, e.g. fog, ground glass, and oiled paper. 
Those substances are opaque that apparently cut off all the 
light-waves and prevent objects from being seen through 
them. When bodies intercept light, they are said to cast 

282. Every point of a luminous body an independent source 
of light-waves. — Place a candle flame in the center of a 
darkened room ; each wall and every point of each wall 
becomes illuminated. Place your- 
self in any part of the room, i.e. 
in any direction from the flame ; 
you are able to see not only the 
flame, but every point of the 
flame ; hence every point of the 
flame must emit light-waves in 
every direction. Every point of 
a luminous body is an independent 
source of light-iuaves, and emits 
them in every direction. Such a 
point is called a luminous point. 

In Figure 225 there are represented a few of the infinite 
number of pencils of light emitted by three luminous points 



of a candle flame. Every point of an illuminated object ab 
receives light from every luminous point. 
283. Images formed through small apertures. 

Exjperiment 1. — Cut a hole about 8^™ square in one side of a box ; 
cover the hole with tin-foil, and prick a hole in the foil with a pin. 
Place the box in a darkened room, and a candle flame in the box near 
the pin hole. Hold an oiled-paper screen before the hole in the foil ; an 
inverted image of the candle flame will appear upon the translucent 
paper. An image is a kind of picture of an object. 

If light-waves from objects illuminated by the sun — e.g. 
trees, houses, clouds, or even an entire landscape — be allowed 
to pass through a small aperture in a window shutter and strike 
a white screen (or a white wall) in a dark room, inverted images 
of the objects in their true colors will appear upon the screen. 
The cause of these phenomena is easily understood. When no 

screen intervenes between the 
candle and the screen A (Fig. 
226), every point of the screen 
receives light from every point 
of the candle ; consequently, 
at every point on A, images of 
the infinite number of points 
of the candle are formed. The 
result of the confusion of 
images is that no image is distinguishable. But let the screen 
B, containing a small hole, be interposed ; then, since light 
travels only in straight lines, the point Y' can receive an 
image only of the point Y, the point Z' only of the point Z, 
and so for intermediate points ; hence a distinct image of the 
object must be formed on the screen A. That an image may 
he distinct, the images of different jjoints of the object must not 
mix, and therefore all rays from each point on the object must 
he carried to the corresponding point on the image. 














■ — ' 


Fig. 226. 


The brightness of the image decreases as the opening is 
made smaller, since less light can pass through it. The 
aperture, if small, may have any shape without affecting 
the outline of the image. The image of the sun is a circle, 
irrespective of the shape of the aperture, if its rays strike 
the screen perpendicularly ; but elliptical, if they strike the 
screen obliquely. 

284. Shadows. 

Experiment 2. — Procure two pieces of tin or cardboard, one 18^™ 
square, the other S^m square. Place the first between a white wall and 
a candle flame in a darkened room. The opaque tin intercepts the light 
that strikes it, and thereby excludes light from a space behind it. 

This space is called a shadow. That portion of the surface 
of the wall that is darkened is a section of the shadow, and 
represents in form a cross section of the body that intercepts 
the light. A section of a shadow is frequently for conven- 
ience called a shadow. Notice that the shadow is made up 
of two distinct parts, — a dark center bordered on all sides 
by a much lighter fringe. The dark center is called the 
umbra., and the lighter envelope is called the penumbra. 

Experiment 3. — Carry the tin nearer the wall, and notice that the 
penumbra gradually disappears and the outline of the umbra becomes 
more distinct. Employ two candle flames, a little distance apart, and 
notice that two shadows are produced. Move the tin toward the wall, 
and the two shadows approach each other, then touch, and finally over- 
lap. Notice that where they overlap the shadow is deepest. This part 
gets no light from either flame, and is the umbra ; while the remaining 
portion gets light from one or the other, and is the penumbra. 

Just so the umbra of every shadow is the part that gets no 
light from the luminous body, while the penumbra is the part 
that gets light from some portion of the body, but not from the 



Experiment 4. — Repeat the above experiments, employing the smaller 
piece of tin, and note all differences in phenomena that occur. Hold a 
hair in the sunlight, about a centimeter in front of a fly-leaf of this book, 

and observe the shadow cast by the 
P — ^ ""^ hair. Then gradually increase the dis- 

tance between the hair and the leaf, 
and note the change of phenomena. 

If the source of light were a single 
luminous point, as A (Fig. 227), the 
shadow of an opaque body B would be of infinite length, and would con- 
sist only of an umbra. But, if the source of light have a sensible size, 
the opaque body will intercept just as many separate pencils of light as 
there are luminous points, and consequently will cast an equal number of 
independent shadows. 

Let A B (Fig. 228) represent a luminous body, and C D an opaque 
body. The pencil from the luminous point A will be intercepted between 

Fig. 227. 

Fig. 228. 

the lines C F and D G, and the pencil from B will be intercepted between 
the lines C E and D F. Hence, the light will be wholly excluded only 
from the space between the lines C F and D F, which enclose the umbra. 
The enveloping penumbra, a section of which is included between the 
lines C E and C F, and between D F and D G, receives light from certain 
points of the luminous body, but not from all. 



1. Why are images formed through apertures inverted ? 

2. Why is the size of the image dependent on the distance of the 
screen from the aperture ? 

3. Why does an image become dimmer as it becomes larger ? 

4. Why do we not imprint an image of our person on every object in 
front of which we stand ? 

5. Upon what fact does a gunner rely in taking sight ? 

6. Explain the umbra and penumbra cast by the opaque body H I, 
Fig. 228. 

7. When will a transverse section of the umbra of an opaque body be 
larger than the object itself ? 

8. When has an umbra a limited length ? 

9. Wbat is the shape of the umbra cast by the sphere C D, Eig. 228 ? 

10. If C D should become the luminous body, and A B a non-luminous 
opaque body, what changes would occur in the umbra and the shadow 

11. Wliy is it difficult to determine the exact point on the ground 
where the umbra of a church-steeple terminates ? 

12. What is the shape of a section of the shadow cast by a circular 
disk placed obliquely between a luminous body and a screen ? What is 
its shape when the disk is placed edgewise ? 

13. The section of the earth's umbra on the moon in an eclipse always 
has a circular outline. What does this show respecting the shape of the 
earth ? 

14. Describe the shadow cast by the earth. 

15. Wliy does the electric arc lamp cast well defined shadows ? 

Section III. 


285. Light requires time to pass through space. — That light 
travels with finite speed was first established in 1676 by the 
Danish astronomer Olaf Eoemer, then engaged in Paris in 
observing the eclipses of Jupiter's moons. He made obser- 


vations on a certain one of Jupiter's satellites wMch. revolves 
round this planet as the moon does round the earth. At 
regular intervals the satellite ^oasses behind the planet 
and is eclipsed within its shadow. The observed intervals, 
however, were found to be shorter than the mean value 
when the Earth and Jupiter were approaching each other, 
and longer when they were receding from each other. It 
was evident that this difference was due to the time con- 
sumed by the light in crossing the intervening spaces. 
From the results of these observations it was calculated 
that light required 16 minutes and 36 seconds to traverse the 
diameter of the earth's orbit, approximately 185,000,000 

It was then an easy matter for Eoemer to determine how 
far light travels per second. The speed of light as deter- 
mined by Eoemer is 192,500 miles per second. It has been 
determined by later experiments and more reliable methods 
that this estimate is too great. The result obtained by 
Michelson at Cleveland (1882) is 299,853 kilometers (= about 
186,380 miles) per second. This may be accepted as probably 
the nearest approximation yet made to the true speed of light 
in a vacuum. At this rate, light would encircle our earth 
between seven and eight times in a second. 

Sound creeps along at the comparatively slow pace of about 
one-fifth of a mile (or -J- Km) per second. The former is the 
speed with which waves in ether are transmitted ; the latter, 
the speed with which waves in air move forward. This great 
difference can be accounted for only on the supposition that 
the ether is far less dense and much more elastic than air. 

Notwithstanding its great speed, light requires no less 
than three years to reach us from the nearest fixed star 
(a Centauri), and from those more distant it requires cen- 
turies. It is thus possible, through the instrumentality of 
light, faintly to conceive of the vastness of space. 


Section IY. 

intensity of illumination. 

286. Unit of measurevient. — The unit generally employed 
for the measurement of the intensity of the light emitted by 
a luminous body is the British candle poiver} It is the in- 
tensity of light emitted by a sperm candle -J in. in diameter, 
burning 120 grains to the hour. 

287. Diminution of intensity of illuminating capacity ivith 
distance. A2)2^lication of the law of inverse squares to light. 
— Light diminishes in intensity, and hence in its power to 
illuminate objects which it strikes, as it recedes from its 
source. The intensity of light diminishes as the square of the 
distance from its source increases. Calling the quantity of 
light falling upon a visiting card at a distance of 2 feet from 
a lamp flame 1, the quantity falling upon the same card at a 
distance of 4 feet is \, at a distance of 6 feet it is i, and so 
on. This is the meaning of the law of inverse squares, as 
applied to light. 

1 The French unit is the carcel (the name given to a lamp), which is equal to 9J 
candles. The unit adopted at the International Congress of Electricians in 1884 is 
the light emitted by a square centimeter of molten platinum at the temperature of 
solidification, or about 2.08 carcels, or 19.8 candles. This is called the platinum 
standard ; and the method, the Yiolle method. Subsequently (1889), o^(j of this unit 
was adopted as the practical standard. 

" By photometric methods it is found that the sun gives us 1575 billions of billions 
times as much light as a standard candle would do at that distance. 

" The intensity of sunlight, or the intrinsic brightness of the sun's surface, is 
quite a different. matter from the total quantity of its light expressed in candle 
power. By intensity we mean the amount of light per square unit of luminous 
surface. From the best data we can get we find that the sun's surface is about 
190,000 times as bright as that of a candle flame ; and about 150 times as bright as 
the lime of a calcium light. 

" The brightest part of an electric arc comes nearer sunlight in intensity than 
anything else that we know, being from one-half to one-quarter as bright as the solar 
surface itself."— Young's Elements of Astronomy. 

" If there were an electric light of 2000-candle power on each square foot of the 
surface of the earth, the whole light from the earth would be less than one billionth 
that from the sun." — Lajstgley. The earth intercepts an extremely small part of 
the whole quantity of light emitted by the sun. 



Fig. 229. 

This law may be illustrated thus : A square card placed 
(say) 1 foot from a certain point in a candle flame, as at A 
(Fig. 229), receives from this point a 
certain quantity of light. The same 
light if not intercepted would go on to 
B, at a distance of 2 feet, and would 
there illuminate four squares, each of 
the size of the card, and being spread 
over four times the area can illuminate 
each square with only one fourth the intensity. If allowed 
to proceed to C, 3 feet distant, it illuminates nine such 
squares, and has but one ninth its intensity at A. The law 
is strictly true only when distance from individual points is 

288. Fhotometry. — The law just established enables us to 
compare the illuminating power of one light with that of an- 
other, and to express by numbers their relative illuminating 
powers. The process is called ijhotometry (light-measuring) ; 
and the instrument employed, 2^ fliotometer. 













II \/ 1 






'^0 '" 

60 ' -i 






Fig. 230. 

289. The Bunsen photometer (Fig. 230) has a screen of 
paper, S, mounted in a box, B, open in front and at the 
two ends. The box slides on a graduated bar. The screen 
has a circular central spot saturated with paraffine, which 
renders the spot more translucent than other portions of the 
screen. One side of the screen is illuminated by the light, 
L, whose intensity is to be measured, and the other side by 



a standard candle, L'. When the screen is so placed that 
the two sides are equally illuminated by the two lights, the 
paraffined spot becomes nearly invisible. When one side is 
more strongly illuminated than the other, the spot appears 
dark on that side and light on the other. The candle power 
of the two lights is directly proportional to the square of their 
respective distances from 

the screen when it is 
equally illuminated on 
both sides. 

In order to render both 
sides of the disk simul- 
taneously visible, two 
mirrors, m and m' (Fig. 

Fig. 231. 

231), are placed in the box in a vertical position so as to 
reflect images of the circular spot in the screen, S, to the 
eyes at E Ei. 


- 1. Suppose that a lighted candle is placed in the center of each of 
three cubical rooms, respectively 10, 20, and .30 feet on a side ; would a 
single wall of the first room receive more light than a single wall of either 
of the other rooms, or less ? 

2. Would one square foot of a wall of the third room receive as much 
light as would be received by one square foot of a wall of the first room ? 
If not, what difference would there be, and why the difference ? 

3. If a board 10 cm square be placed 25 cm from a candle flame, the 
area of the shadow of the board cast on a screen 75 cm distant from the 
candle will be how many times the area of the board ? Then the light 
intercepted by the board will illuminate how much of the surface of the 
screen if the board be withdrawn ? 

4. Give a reason for the law of inverse squares. 

5. To what besides light has this law been found applicable ? 

6. The two sides of a paper disk are illuminated equally by a candle 
flame 50 cm distant on one side and a gas flame 200 cm distant on the 
other side. a. Compare the intensities of the two lights at equal dis- 
tances from their sources. 6. If the candle be a standard candle, what 
is the intensity of the gas flame ? 


Section Y. 

apparent size of an object. 

290. Visual angle. 

Experiment. — Prick a pin-hole in a card, place an eye near the hole, 
and look at a pin about 20 cm distant. Then bring the pin slowly toward 
the eye, and the dimensions of the pin will appear to increase as the dis- 
tance diminishes. 

Why is this ? We see an object by means of its image 
formed on the retina of the eye ; and its apparent magnitude 
is determined by the extent of the retina covered by its 
image. Rays proceeding from opposite extremities of an 
object, as AB (Fig. 232), meet and cross each other within 


Fig. 232. 

the eye. Now, as the distance between the points of the 
blades of a pair of scissors depends upon the angle that the 
handles form with each other, so the size of the image 
formed on the retina depends upon the size of the angle, 
called the visual angle, formed by these rays as they enter 
the eye. But the size of the visual angle diminishes approxi- 
mately as the distance of the object from the eye increases, 
as shown in the diagram ; e.g. at twice the distance the angle 
is about one-half as great ; at three times the distance the 
angle is one-third as great ;and so on. Hence, distance affects 
the aioparent size of an object. Our judgment of the size of 
objects is, however, influenced by other things besides the 
visual angle which they subtend. 



Section VI. 


291. Miry^oj's. Images. — Objects having polished surfaces 
which reflect light regularly (i.e. do not scatter the light), 
and show images of objects presented to them, are called 
mirrors. The mirror itself, if clean and smooth, is scarcely 
visible. An image is a picture of an object. According to 
their shape mirrors are called planey concave^ convex, spherical, 
parabolic, etc. 

Experiment 1. — a. Look at the mirror M through the hole marked O 
in the metal band (Fig. 233). You see in the mirror an image of the 
hole through which you look, 
but you do not see the image 
of any of the other holes. Eays 
that pass through this hole 

strike the mirror perpendicu- 
larly and are said to be normal 
to the mirror. Rays falling 
upon an object are called inci- 
dent rays. The point where a ray strikes is called the point of incidence. 
The reflected rays in this case are thrown back in the same lines and 
through the same hole that the incident rays travel. Bays normal to a 
mirror after reflection simply retrace their own course, h. Next hold a 
candle flame at one of the other holes, e.g. at the hole marked 10. You 
can see the image of the candle flame only through the hole of the same 
number and at an equal distance on the other side. The angle which an 
incident ray makes with a line normal at the point of incidence is called 
the angle of incidence, and the angle made by a reflected ray with the 
normal is called the angle of reflection. 

Law of Reflection. The angles of incidence and reflec- 
tion are in the same plane, and are equal. 
292. The luave-theory applied to reflection. 

The following is an explanation of reflection in accordance with 
the wave-theory. Suppose KA, ND, etc. (Fig. 234) to be parallel 
rays of a beam of light falling on a plane mirror HI. KLMN 



may represent the plane front of one of the waves. As soon as the 
wave reaches A, that point becomes the origin of a disturbance in 
the ether, which spreads out in the form of a sphere having its 
center at A. This disturbance may for convenience be called an 
undulation. Let the arc of a circle described around A as a center 
denote the boundary which the undulation has reached during the 
interval between the arrival of the plane-wave at A and D respec- 
tively ; then the radius of this circle is equal to the excess of ND 

over KA, because so long as light travels in the same isotropic 
medium its speed in all directions is the same. Similarly let circles 
be described around points B, C, and J) with radii determined in 
the same manner. A straight line a&cD, drawn tangent to these 
circles at the points a, 6, etc. , represents a plane reflected wave-front 
corresponding to the plane incident- wave. It is inclined to H I at 
the same angle but in the opposite direction from a normal. 

293. The doubled angle of reflection. 

When a mirror is rotated, a beam 
of light reflected from it is deflected 
through an angle equal to twice that 
of the rotation of the mirror. In Fig. 
2.35, I M is an incident ray, M R a 
reflected ray. If the mirror be turned 
into the position A'B', the reflected 
ray is now M R' ; the reflected ray 
has moved through the angle R M R', 
which is equal to twice the angle 
A M A'. This fact suggests an invalu- 
able method of making minute motions 
apparent. The reflected ray itself serving as a weightless index- 
pointer of any desired length, and capable of magnifying motions to 


any desired extent. For example, the almost imperceptible motion 
of the pulse may be made visible to a large audience in the follow- 
ing manner : Lay (or 
stick with wax) a tiny K 
mirror upon the throb- 
bing part of the wrist. 
In a darkened room 
project from a lantern 
(or porte-lumifere) a 
small beam of light 

obliquely upon the ^ig 236 

mirror M (Fig. 236), 

and let the reflected beam strike the ceiling above. The spot of light 
on the screen will move several inches with each pulsation. 

294. Intensity of reflected light. — The intensity of reflected 
light increases with the polish of the reflecting surface and 
with the obliquity of the incident rays. It also depends 
largely upon the nature of the medium from which it is 
reflected. For example, at perpendicular incidence, water 
reflects about the fiftieth part of the incident light while 
mercury reflects about two-thirds ; but at an incidence of 
89|-° each reflects about 72 per cent of the incident light. 
The varnished surfaces of furniture appear much brighter 
when viewed obliquely than when seen by light from a win- 
dow reflected less obliquely. Light reflected from the surface 
of a pond just before the sun sets is much more dazzling than 
at noon when the sun is overhead. This is due in part to 
the fact that we are in a suitable position to observe it. 

295. Diffused light. 

Experiment 2. — Introduce a small beam of light into a darkened room, 
by means of a porte-lumiere, and place in its path a mirror. The light is 
reflected in a definite direction. If the eye be placed so as to receive the 
reflected light, it will se'e, not the mirror, but the image of the sun, and 
the light will be painfully intense. Substitute for the mirror a piece of 
unglazed paper. The light is not reflected by the paper in any definite 
direction, but is scattered in every direction, illuminating objects in the 



vicinity and rendering them visible. Looking at the paper, you see, not 
an image of the sun, hut the paper, and you may see it equally well in 
all directions. 

The dull surface of the paper receives light in a definite 
direction, but reflects it in every direction ; in other words, 
it scatters or diffuses the light. The difference in the phe- 
nomena in the two cases is caused by the difference in the 
smoothness of the two reflecting surfaces. AB (Fig. 237) 
represents a smooth surface, like that of glass, which reflects 
nearly all the rays of light in the same direction, because 
nearly all the points of reflection are in the same plane. 
C D represents a surface of paper having the roughness of 
its surface greatly exaggerated. The various points of re- 

FlG. 237. 

flection are turned in every possible direction ; consequently, 
light is reflected in every direction. Thus, the dull surfaces 
of various objects around us reflect light in all directions, 
and are consequently visible from every side. Objects ren- 
dered visible by reflected light are said to be illum.inated. 

By means of regularly reflected light we see images of 
objects in mirrors, but only from definite positions, i.e. in 
definite directions ; by means of diffused light we see the 
object itself from every direction. Whether we see the image 
of the source of the light (the eye being situated so as to 
receive the regularly reflected light), or the object on which 
the light falls, or both at the same time, depends largely 
upon the degree of smoothness possessed by the surface that 
reflects the light. Polished metals are the best mirrors. 
Surfaces of liquids at rest are excellent mirrors. It is some- 



Fig. 238. 

times difficult to see a smooth, surface of a pond surrounded 
by trees and overhung by clouds, as the eye is occupied by 
the reflected images of these objects ; but a faint breath of 
wind, slightly rippling the surface, will reveal the water. 

296. Reflection fi^om jplane mirrors ; virtual images. — MM 
(Fig. 238) represents a plane mirror, and AB a pencil of 
divergent rays proceeding from the 
point A of an object A H. By erect- 
ing perpendiculars at the points of 
incidence, or the points where these 
rays strike the mirror, and making 
the angles of reflection equal to the 
angles of incidence, the paths B C and 
E C of the reflected rays are found. 

Every visible point of an object 
sends a cone of rays to the eye. The 
point always appears at the place 
whence these rays seem to emerge, i.e. at the real apex of 
the cone. If the direction of these rays be changed by re- 
flection, or in any other manner, the point will appear to be 
in the direction of the rays as they enter the eye ; thus the 
point A appears to lie in the direction C D ; and the point H, 
in the direction G N. The exact location of these points may 
be found by continuing the rays CB and CE behind the 
mirror, till they meet at the points D and N. Thus, the 
pencils E C and B C appear to emanate from the points N 
and D ; and the whole body of light-waves received by the 
eye seems to come from an apparent object ND behind the 
mirror. This apparent object is called an image. An image 
is a point or a series of points from which a diverging pencil 
of rays comes or appears to come. As of course no real 
image can be formed back of a mirror, such an image is 
called a virtual or an imaginary image. It will be seen, by 
construction, that an image in a plane mirror appears as far 


behind the mirror as the object is in front of it, and is of the 
same size and shape as the object: 

It appears from the above diagram that divergent incident 
rays remain divergent after reflection from a plane mirror. 
In a like manner the student may construct diagrams, and 
show that parallel incident rays are parallel after reflection, 
and convergent incident rays are convergent after reflection, 
i.e. reflection from, a plane mirror does not change the angle 
between rays. 

297. Reversion of images. — When we look at our own 
faces in a mirror we discover a lateral reversion. The right 
cheek is the left cheek in the image ; the hair parted on the 
left is parted on the right in the image. 

If the mirror be vertical, objects appear in their proper 
relations to the horizon ; but, if the mirror have any other 
position, objects assume unnatural postures. Thus, turn this 
book so that the mirror M M (Fig. 238) may represent a hori- 
zontal mirror, and AH a vertical object above it, and it will 
be seen that the image appears inverted. To verify this, 
place a mirror in a horizontal position, and set on it a goblet 
of water. The image of the goblet will appear upside down. 
In a mirror inclined at an angle of 45° to the horizon, the 
image of an erect object appears horizontal, while the image 
of a horizontal object appears erect. 

298. Multiple reflection; images of images. 

"When light is reflected successively from two plane mirrors, the 
image in the first becomes the object for the second mirror, and 
the second image is found in precisely the same manner as the first 
one. Again, the second image serves as an object for a third image, 
and so on. If the two mirrors be parallel, as A and B (Fig. 239), 
the series of images, theoretically infinite in number, is formed on 
a common straight line normal to the mirrors and at regularly in- 
creasing distances from the mirrors. Thus a' is the primary image 
of the object a in mirror A (to avoid confusion, a pencil from only 
one point o is drawn). The light reflected at c enters the eye as 



though it came from o\ Other rays reflected from A at e diverge 
as though they emanated from a', and are reflected from B at e\ 
and may be regarded as proceeding from a real object at a', whose 
image is 6, as far back of B as a' is in front of B. The light re- 
flected from B to A again diverges as though it really came from 6, 
and h being regarded as a real object, its image would be formed 
at a'', and the pencil which enters the eye seems to proceed from 

o''% having been reflected at e'' as though it came from o'\ The 
pencil which would enter the eye from a third image at the left of 
a" may be traced through all its reflections in like manner. As 
some light is lost at each reflection, the images decrease in bright- 
ness as they recede. 

A kaleidoscope is constructed on the principle of multiple re- 
flection. It consists of a tube containing three mirrors placed at 
angles of 60°. Pieces of colored glass, free to move at one end of 
the tube, are seen through an eye-piece 
at the opposite end of the tube, multi- 
plied by repeated reflections. 

Multiplied images of a small, bright 
object, as of a candle flame (Fig. 240), 
often seen in a glass mirror, are pro- 
duced by repeated reflections between 
the anterior surface and the silvered 
posterior surface of the mirror. At 
each internal impact on the first sur- 
face some light escapes, and shows us 
an image, while another portion is re- 
flected to the back, and thence forward 
again, showing another image, and so on. Fig. 240. 

Fig. 241. 


299. Reflection from concave mirrors. — ^Let MM' (Fig. 241) 
represent a section of a concave spherical mirror, which may 
be regarded as a small part of a hollow spherical shell having 

a polished interior 
surface. The distance 
MM' is called the di- 
ameter of the mirror. 
C is the center of the 
sphere, and is called 
the center of curvature. 
G is the vertex of the 
mirror. A straight line DG drawn through the center of 
curvature and the vertex is called the jprinci'pal axis of the 
mirror. A concave mirror may be considered as made up of 
an infinite number of small plane surfaces. All radii of the 
mirror, as CA, CG, and CB, are perpendicular to the small 
planes which they strike. If C be a luminous point, it is 
evident that all light-waves emanating from this point, and 
striking the mirror, will be reflected to their source at C. 

Let E be any luminous point in front of a concave mirror. 
To find the direction that rays emanating from this point 
take after reflection, draw any two lines from this point, as 
EA and EB^ representing two of the infinite number of rays 
composing the divergent pencil that^trikes the mirror. ISText, 
draw radii to the points of incidence A and B, and draw the 
lines AF and BF, making the angles of reflection equal to 
the angles of incidence. Place arrow-heads on the lines rep- 
resenting rays to indicate the direction of the motion. The 
lines AF and BF represent the direction of the rays after 

It will be seen that the rays after reflection are convergent, 
and meet at the point F, called the focus. This point is the 
focus of all reflected rays that emanate from the point E. It 
is obvious that if F were the luminous point, the lines AE 


and B E would represent the reflected rays, and E would be 
the focus of these rays. Since the relation between the two 
points is such that light-waves emanating from either one are 
brought by reflection to a focus at the other, these points are 
called conjugate foci. Conjugate foci are two points so related 
that the image of either is formed at the other. The rays EA 
and E B, emanating from E, are less divergent than rays F A 
and FB, emanating from a point F less distant from the 
mirror, and striking the same points. Bays emanating from 
D, and striking the same points A and B, will be still less 
divergent ; and if the point D were removed to a distance of 
many miles, the rays incident at these points would be very 
nearly parallel. Hence rays may be regarded as practically 
parallel when their source is at a very great distance, e.g. the 
sun's rays. If a sunbeam, consisting of a bundle of parallel 
rays, as E A, D G, and HB (Fig. 242), strike a concave mirror 
in a direction parallel with its principal axis, 
these rays become convergent by reflection, 
and meet at a point (F) in the principal 
axis. This point, called the principal focus, 
is about halfway between the center of curva- 
ture and the vertex of the mirror. 

On the other hand, it is obvious that di- 
vergent rays emanating from the principal focus of a concave^ 
mirror become parallel by reflection. 

If a small piece of paper be placed at the principal focus 
of a concave mirror, and the mirror be exposed to the parallel 
rays of the sun, the paper will quickly burn. 

Construct a diagram, and show that rays proceeding from a 
point between the principal focus and the mirror are divergent 
after reflection, but less divergent than the incident rays. On 
reversing the direction of the rays, the same diagram will 
show that convergent rays are rendered more convergent by 
reflection from concave mirrors. 



\^^ ' H 
, 11 


Fig. 242. 



The general effect of a concave mirror is to inci^ease the con- 
vergence or to decrease the divergence of incident rays. 

300. Spherical aberration of mirrors. 

The statement that parallel rays after reflection from a concave 
mirror meet at the principal focus is only approximately true. It 
is strictly true only of parabolic mirrors such as are used in the 
head-lights of locomotives. Consequently parabolic mirrors are 
used when it is desired to bring the plane-fronted light-waves of a 
distant star accurately to a focus, or to change a divergent pencil 
to a parallel beam ; in the latter case the source of light is placed at 
the focus of the paraboloid. In spherical mirrors when the pencil 
is broad, the outside rays or those which are incident upon the 
mirror farthest from its vertex are brought to a focus nearer the 
mirror than the inner rays ; consequently the image furnished by 
a luminous point is a circle brightest toward its center. This phe- 
nomenon is called the spherical aberration of a mirror. It renders 
the definition of the images of objects, especially of broad objects, 

very bad. In conse- 
quence of this it is often 
necessary to cut off the 
outside rays by a dia- 
I)hragm, which improves 
the definition at the ex- 
pense of the brightness 
of the image. 

By constructing a 
number of rays (Fig. 
243) we may show that 
all rays after reflection 
are tangent to a charac- 
teristic curve called a 
caustic. The light emit- 
ted from a single point, as A, is spread over the surface produced 
.by the revolution of the line ST about the axis AM. This curve 
formed in milk by reflection from the interior surface of a bright 
tin pail is commonly called " the cow's foot." 

301. Formation of images. — To determine the position 
and kind of images formed in concave mirrors of objects 

Fig. 243. 



Fig. 244, 

placed in front of them, proceed as follows : Locate the ob- 
ject, as DE (Fig. 244). Draw lines, E A and DB, from the 
extremities of the object through 
the center of curvature of the mir- 
ror, to meet the mirror. These lines 
are called secondary axes. Incident 
rays along these lines will return 
by the same paths after reflection. 
Draw another line from D to any 
point in the mirror, e.g. to F, to rep- 
resent another of the infinite num- 
ber of rays emanating from D. Make the angle of reflection 
CFD' equal to the angle of incidence CFD, and the reflected 
ray will intersect the secondary axis D B at the point D'. 
This point is the conjugate focus of all rays proceeding from 
D. Consequently, an image of the point D is formed at D'. 
This image is called a real image, because rays actually meet 
at this point. In a similar manner, find the point E', the 
conjugate focus of the point E. The images of intermediate 

points between D and E lie 
between the points D' and E' ; 
and, consequently, the image 
of the object lies between 
those points as extremities. 

If, for the second ray to be 
drawn from any point, we 
select that ray which is par- 
allel with the principal axis, 
as AG (Fig. 245), it will not be necessary to measure angles. 
For this ray, after reflection, must pass through the principal 
focus F ; and consequently the conjugate focus A' is easily 
found, and so for the point B' and intermediate points. Both 
methods of constructing images should be practiced by the 

Fig. 245. 



Fig. 246. 

It thus appears that an image of an object placed beyond the 
center of curvature of a concave TYiirror is real, inverted, smaller 
than the object, and located between the center of curvature and 
the principal focus of the mirror. An eye placed in a suitable 
position to receive the light, as at H (Fig. 246), will receive 

the same impression from the 
reflected rays as if the image 
E'D' were a real object. For 
a cone of rays originally ema- 
nates from (say) the point D of 
the object, but it enters the 
eye as if emanating from D', 
and consequently appears to 
originate from the latter point. A person standing in front 
of such a mirror, at a distance greater than its radius of 
curvature, will see an image- of himself suspended, as it were, 
in mid-air. Or, if in a darkened room an illuminated object 
be placed in front of the mirror, and a small oiled-paper screen 
be placed where the image is formed, a large audience may 
see the image projected upon the screen. 

If E'D' (Fig. 246) be taken as the object, then the direc- 
tion of the light in the diagram will be reversed, and ED 
will represent the image. 
Hence, the image of an 
object placed between the 
principal focus and the 
center of curvature is also 
real and inverted, hut 
larger .than the object, and 
located beyond the center 
of curvature. The image in this case may be projected upon 
a screen, but it will not be so bright as in the former case, 
because the light is spread over a larger surface. 

Construct an image of an object placed between the principal 

Fig. 247. 




Fig. 248. 

focus and the mirror, as in Fig. 247. It will be seen in 

this case that a pencil of rays proceeding from any point of 

an object, e.(j. D, has no actual focus, but 

appears to proceed from a virtual focus 

D', back of the mirror, and so with other 

points, as E. The image of an object 

placed between the priiicipal focus aiid 

the mirror is virtual, erect, larger than 

the object, and bach of the mirror. 

The diagram in Fig. 248 suggests the 
method of finding the disposition of a pencil of rays emanating 
from any point {e.g. A) after reflection from a convex mirror. 
Construct an image of an object placed in front of a convex 

302. Illustrative experiments. 

Experiment 3. — Hold some object, e.g. a rose, as ah (Fig. 249), a few 
feet in front of a concave mirror. Looking in the direction of the axis 

of the mirror you see a 
small inverted image, 
AB, of the object, be- 
tween the center of cur- 
vature, C, of the mirror 
and its principal focus, 

Evidently if AB rep- 
resent an object placed 
between the principal 
focus and the center of 
curvature, then ah will 
represent the image of 
the object. 
Experiment 4- — Place a candle in an otherwise dark room 20 feet 
from the mirror, catch the focused light-waves upon a paper screen, and 
show that the focus is about half-way between the vertex and the center 
of curvature of the mirror. 

Experiment 5. — Advance the distant candle flame toward the mirror, 
moving it up and down. (1) Show that the focus advances to meet the 

Fig. 249. 


flame, and that when the flame is raised the focus is depressed, and the 
converse. (2) Show that when the flame is at the center of curvature, 
the focus is also there. (3) Show that when the flame is between the 
center of curvature and the principal focus, tlie focus of the flame is 
farther away than the center of curvature. (4) Show that when the 
flame is at the principal focus, the reflected rays are parallel, or the focus 
is at an infinite distance. (5) Show that when the flame is still nearer, 
the reflected rays diverge and appear to come from a point behind the 
mirror. (6) Notice that in all cases except the last the images are real 
and inverted, and that in all cases where a real image is formed, the' 
flame and the image may change places. 

Experiment 6. — Form a real image of the flame between yourself and 
the mirror; view the image through a convex lens (§ 318); show that 
the image can be magnified by a convex lens, and thereby illustrate the 
principle of the astronomical reflecting telescope (§ 388). 

When light emitted by a luminous point at a distance / is reflected by 
a concave spherical mirror, it is reflected back to an approximate focus 
at a distance /'. The relation between the distance of the source /, the 
distance of the focus /', and the radius of the mirror r, is expressed by 
the following simple formula ^ : 

,., yt 


From this 


we get 

(2) r=-^ 


which gives the distance of the image from the mirror in terms of / and r. 

(a) Since the sum of the reciprocals of / and/' is a constant, -, it fol- 


lows that as/ increases/' decreases, and when/ becomes infinite/' = -• 

Hence parallel rays {i.e. rays from an infinitely distant source) come to a 
focus at a point half way between a mirror and its center of curvature. 


When/ = -, /'= CO ; i.e. rays emanating from the principal focus become 

parallel, and the waves plane-fronted. 

(6) When / decreases /' increases, i.e. the object and image approach 

2 2 
each other. When / equals /', ^= -, i-e. object and image coincide at 

the center of curvature. 

1 See Ganot, p. 434 ; Barker, p. 416. 


r 1 2 

(c) When / is less than - , - is greater than - , and /' is therefore 
2 / r 

negative, and the image is behind the mirror, and hence virtual. Dis- 
tances in front of the mirror are considered positive, and those back of 
the mirror negative. 


1. When an object is located at a distance from a concave mirror 
equal to its radius, will any image be formed ? Why ? 

2. What is the effect of placing the object at the principal focus? 

3. a. When is the real image formed by a concave mirror smaller 
than the object ? b. When is it larger ? 

4. a. When is the image formed by a concave mirror real ? b. When 
is it virtual ? 

5. a. Is the image of an object formed by a convex mirror real or 
virtual ? b. Is it larger or smaller than the object ? c. Is it erect or 
inverted ? 

6. Is the general effect of a convex mirror to collect or to scatter rays ? 

7. The radius of a concave spherical mirror is 20 inches. Determine 
the conjugate focus for a point on the principal axis 15 inches from the 

8. Why do images formed by a surface of v^ater appear inverted ? 

9. a. What kind of wave-front has a beam of parallel rays ? b. What 
change in front of such a beam occurs when it strikes each of the follow- 
ing mirrors : viz. a plane, a concave, and a convex mirror ? 

10. Where is the conjugate focus of light emanating from each of the 
following points : a. the center of curvature of a concave spherical 
mirror ; b. a point on the principal axis at an infinite distance from the 
mirror ; c. a point on the principal axis beyond the center of curvature 
and at a finite distance from the mirror ; d. a point on the principal axis 
between the center of curvature and the principal focus ; e. a point be- 
tween the principal focus and the vertex ? 

11. How could you find the radius of curvature of a concave spherical 
mirror by optical means alone ? 



Section VI. 


303. Introductory experiments. 

Experiment 1. — Into a darkened room admit a sunbeam so that its 

rays may fall obliquely on the bottom of the basin (Fig. 250), and note 

the place on the bottom where the edge of the shadow D E cast by the 

side of the basin D C meets the bottom 
at E. Then, without moving the basin, 
fill it evenly full with water sliglitly 
clouded with milk or with a few drops 
of a solution of mastic in alcohol. It 
will be found that the edge of the shadow 
has moved from D E to D F, and meets 
the bottom at F. Beat a blackboard 
rubber, and create a cloud of dust in 
the path of the beam in the air, and you 
will discover that the rays GD that 
graze the edge of the basin at D become 
bent at the point where they enter the 

water, and now move in the bent line G D F, instead of, as formerly, in 

the straight line GDE. The path of the line in the water is now nearer 

to the vertical side D C ; in other words, this part of the beam is more 

nearly vertical tha^n before. 

Experiment 2. — Place a coin (A, Fig. 251) on the bottom of an empty 

basin, so that, as you look through a small hole in a card B C over the 

edge of the vessel, the coin is just out of sight. 

Then, without moving the card or basin, fill the 

latter with water. Now, on looking through 

the aperture in the card, the coin is visible. H 

The beam AE, which formerly moved in the 

straight line AD, is now bent at E, where it 

leaves the water, and, passing through the aper- q 

ture in the card, enters the eye. Observe that 

as the beam passes from the water into the air it 

is turned farther from a vertical line E F ; in other words, the beam is 

farther from the vertical than before. 

Fig. 251. 


Experiment 3. — From the same position as in the last experiment, 
direct the eye to the point G in the basin filled with water. Reach your 
hand around the basin, and place your finger where that point appears 
to be. On examination, it will be found that your finger is considerably 
above the bottom. Hence, the effect of the bending of rays, as they pass 
obliquely out of water, is to cause the bottom to appear more elevated than 
it really is; in other words, to cause the water to appear shallower than 
it is. 

Experiment 4. — Thrust a pencil obliquely into water ; it will appear 
shortened, and bent at the surface of the water, and the immersed portion 
will appear elevated. 

Experiment 5. — Place a piece of wire (Fig. 252) vertically 
in front of the eye, and hold a narrow strip of thick plate glass 
horizontally across the wire, so that the light-waves from the ' | 
wire may pass obliquely through the glass to the eye. The wire 
will appear to be broken at the two edges of the glass, and the 
intervening section will appear to the right or left according 
to the inclination of the glass ; but if the glass be not inclined to the one 
side or the other, the wire does not appear broken. 

When a ray of light passes from one medium into another 
of different density, it is bent or refracted at the interface 
between the two mediums unless it meet this plane perpen- 
dicularly. In the latter case there is no refraction. If it 
pass into an optically denser medium, it is refracted toward 
the perpendicular to this plane ; if into a rarer medium, it is 
refracted from the perpendicular. It is not universally true 
that the denser mediums are the more highly refracting. 
The refractive power of water is less than that of alcohol or 
oil of turpentine. A substance which has a higher refractive 
powder than another is said to be oi)ticallij denser. 

The angle GD (Fig. 250) is called the angle of iiicidence ; 
EDN, the angle of refraction ; and EDF, the angle of devia- 

304. Cause of refraction. — Foucault and others have 
proved by careful experiments that the speed of light is 
much less in water than in air. It is less in glass than in 



water, and much less in diamond than in glass. Every trans- 
parent substance has its own rate of transmission. It would 
seem that there is an interaction between the ether and the 
molecules of matter such that in different mediums the ether- 
waves are unequally retarded. T 
Let the series of parallel lines AB (Fig. 253) represent a 

series of wave-fronts leaving an 
object C, and passing through a 
rectangular piece of glass DE, 
and constituting a beam. Every 
point in a wave-front moves with 
equal velocity as long as it trav- 
erses the same medium ; but the 
point a of a given wave ab enters 
the glass first, and its velocity is 
impeded, while the point b re- 
tains its original velocity; so 
that, while the point a moves to 
a', b moves to b', and the result is that the wave-front assumes 
a new direction (very much in the same manner as a line of 
soldiers executes a wheel), and a ray or a line drawn perpen- 
dicularly through the series of waves is turned out of its 
original direction on entering the glass. Again, the extremity 
c of a given wave-front cd first emerges from the glass, when 
its velocity is immediately quickened ; so that, while d ad- 
vances to d', c advances to c', and the direction of the ray is 
again changed. The direction of the ray after emerging 
from the glass is parallel to its direction before entering it, 
but it has suffered a lateral displacement. 

It is evident that if the ray enter the new medium in a 
direction perpendicular to its surface, i.e. with its wave- 
front parallel to this surface, all parts of the wave-front 
will be retarded simultaneously and no refraction will take 

Fig. 253. 


305. The ivave-theory ajjplied to refraction. 

Let K A, ND, etc. (Fig. 254), be parallel rays of a beam of light 
falling on a plane refracting surface AD. Let KLMN denote the 
plane front of a wave. The wave reaches the refracting surface 
first at A, and then 
successively at other 
points in A B C D. 
As soon as the wave 
reaches A, that point 
becomes the origin 
of an undulation 
in the ether which 
spreads out in all 
directions in the me- 
dium in the form of 
a sphere having its 
center at A. The 
speed with which 
motion is propagated 

in the new medium we suppose to be less than that in the first 
medium. Describe a circle with the center A, and with a radius 
equal to the distance a wave would move in the new medium in the 
same time as it would describe the excess of N D over K A in the 
first medium. Let circles be described from B, C, and other points 
of AD, according to the same law. Then a straight line a&cD 
touching all these circles represents a plane refracted wave. It can 
be demonstrated that the sine (see § 307) of the angle K A G bears 
the same ratio to the sine of the angle a AH as the speed of light 
in the first medium bears to the speed in the new medium, i. e. as 
the excess of N D over K A bears to A a. 

306. Failure of the emission theory to account for the refrac- 
tion of light. 

Fig. 254. 

To explain refraction from a rare to a denser medium according 
to the emission theory of light it is necessary to assume that when 
a light particle shot from a luminous body comes within a very 
small distance of the surface of separation between two mediums, 
it begins to be attracted towards the surface so that its component 



velocity perpendicular to the surface gradually increases till it 
reaches a limited distance on the other side of the surface. That is, 
the speed of light should be by the theory greater in dense than in 
rarer mediums ; whereas the reverse is found to be true. Hence 
the failure of the emission theory to account for the phenomenon 
of refraction. 

307. Index of refraction. — The deviation of light-waves in 
passing from one medium into another, depends upon the 
mediums and the angle of incidence. It diminishes as the 
angle of incidence diminishes, and is zero when the incident 
ray is normal. It is highly important, knowing the angle of 
incidence, to be able to determine the direction which a ray 

will take on entering a new 
medium. Describe a circle 
around the point of inci- 
dence A (Fig. 2ob) as a 
center ; through the same 
point draw IH perpendicu- 
lar to the surfaces of the 
two mediums, and to this 
line drop perpendiculars 
B D and C E from the points 
where the circle cuts the 
ray in the two mediums. 
Then suppose that the per- 
pendicular B D is j% of the radius AB ; now this fraction 
y8g. is called (in trigonometry) the sine of the angle DAB. 
Hence, -^^ is the sine of the angle of incidence. Again, if we 
suppose that the perpendicular CE is f^ of the radius, then 
the fraction -^^ is the sine of the angle of refraction. The 
sines of the two angles are to each other as y8_ ; _6_^ or as 
4:3. The quotient (in this case 1 = 1.33+) obtained by 
dividing the sine of the angle of incidence by the sine of 
the angle of refraction, generally expressed in the form of a 

Fig. 255. 


decimal fraction, is called the iiidex of refraction. It can be 
proved to be the ratio of the velocity of the incident to that of 
the refracted light-waves. 

308. SnelVs ^^Law of Sines. ^^ — We have found that a ray 
of light in passing obliquely from a medium into another of 
different density suffers refraction, and the greater the angle 
of incidence the greater the deflection. Snell in 1621 dis- 
covered the law which governs these variable angles of de- 
flection. It is called the "law of sines": ^^ The incident and 
refracted rays are in the same plane with the normal to the 
surface; they lie on opposite sides of it, and the sines of their 
inclinations hear a constant ratio to each other." The incident 
ray may be more or less oblique, still the index of refraction 
remains the same. 

309. Indices of refraction. — The index of refraction for 
light-waves in passing from air into water is approximately f , 
and from air into glass f ; of course, if the order be reversed, 
the reciprocal of these fractions must be taken as the indices ; 
e.g. from water into air, the index is |; from glass into air, f. 
When a ray passes from a vacuum into a medium, the re- 
fractive index is greater than unity, and is called the absolute 
index of refraction. The relative index of refraction, from any 
medium A into another B, is found hy dividing the absolute 
index of B by the absolute index of A. 

It will be shown later that the refractive index varies with 
wave-length. The following table is intended to represent 
mean indices for light-waves : — 

Table of Absolute Indices. 

Lead chromate 2.97 

Diamond (about) 2.5 

Carbon disulphide 1.64 

Flint glass (about) 1.61 

Agate 1.54 

Canada balsam 1.53 

Crown glass (about) . . . 1.53 

Spirits of turpentine . . . 1.48 

Alcohol 1.37 

Humors of the eye (about) . 1.35 

Pure water . . . . . . . 1.33 

Air at 0° C. and 760"™ pressure 1.000294 



310. Given the direction of the incident ray and the re- 
fractive index, to determine the direction of the refracted ray. — 
Let L (Fig. 256) be the incident ray ; draw a circle with 

the point of incidence, 0, as a center. 
Divide C by the refractive index, and 
set off the quotient, D, on the other 
side of 0. Draw DB perpendicular to 
the surface at D, meeting the circum- 
ference at B; then B is the direction 
of the refracted ray required. For it is 

Fig. 256. apparent that — - is the same as the 

ratio of the sines, and this ratio is by construction equal to 
the refractive index. 

311. Some phenomena of refraction. Refraction into a 
rarer medium. — A stick partly immersed in water appears 
to be bent upwards and shortened unless its position is 
vertical, when the part 
immersed appears sim- 
ply shortened. Fig. 
257 explains this. The 
dotted lines represent 
the real position of the 
submerged part of the 
stick, and dotted lines 
diverging from a point 
at the bottom of the 
stick show the course 
of the rays which reach 
the eye from that point. On reaching the surface they are 
bent from the perpendicular, and the bottom of the stick is 
seen in the direction from which the rays actually enter the eye. 

Viewed obliquely, the depth of water cannot appear greater 
than I its real depth. Hence the shoaling effect of still 

Fig. 257. 



water in which the bottom is visible. To an eye under 
water the surface must appear at least | of its real dis- 

312. Critical angle; total reflection. — Let S S' (Fig. 258) 
represent the boundary surface between two mediums, and 
A and B incident rays in the more refractive medium 
(e.g. glass) ; then D and E may represent the same rays 
respectively after they enter the less refractive medium (e.g. 
air). It will be 
seen that, as the 
angle of incidence 
is increased, the re- 
fracted ray rapidly 
approaches the sur- _ 
face S. Now, | 
there must be an | 
angle of incidence 1 
(e.^. COM) such that | 
the angle of refrac- I 
tion will be 90°; in [ 
this case the inci- 
dent ray CO, after 
refraction, will just graze the surface S. This angle (COM) 
is called the critical or limiting angle. Any incident ray, mak- 
ing a larger angle with the normal than the critical angle, as 
LO, cannot emerge from the medium, and consequently is 
not refracted. Experiment shows that all such rays undergo 
internal reflection ; e.g. the ray L is reflected in the direc- 
tion ON. Reflection in this case is perfect, and hence is 
called total reflection. Total reflection occurs when rays in the 
more refractive medium are incident at an angle greater than 
the critical angle. 

Surfaces of transparent mediums, under these circum- 
stances, constitute the best mirrors possible. The critical 

Fig. 258. 



angle diminishes as the refractive index ^ increases. For 
water it is about 48|-° ; for flint glass, 38° 41'; and for the 
diamond, 23° 41'. Light-waves cannot, therefore, pass out of 
water into air with a greater angle of incidence than 48^°. 
The brilliancy of gems, particularly the diamond, is due in 
part to their extraordinary power of reflection, arising from 
their large indices of refraction. 

313. Illustrations of refraction and total reflection. 

Experiment 6. — Observe the image of a candle flame reflected by the 
surface of water in a glass beaker, as in Fig. 259. 

Experiment 7. — Thrust the closed end of a glass test-tube (Fig. 260) 
into water, and incline the tube. Look down upon the immersed part 

Fig. 259. 

Fig. 260. 

of the tube, and its upper surface will look like burnished silver, or as if 
the tube contained mercury. Fill the test-tube with water, and immerse 
as before ; the total reflection which before occurred at the surface of the 
air in the submerged tube now disappears. Explain. 

1 It must be evident on inspection of Fig. 258 that a ray traveling in the direc- 
tion SO will be refracted in the direction OC. The angle SOK is a right angle, 
and the sine of a right angle = 1. Therefore the index of refraction of the medium 

^=— : Thus, to get the index of refraction of any substance it is only 

sm crit. ang. 

necessary to find the critical angle of the substance. This principle has been applied 

by Kohlrausch in his total reflectometer, in determining indices of refraction of 





Fig. 261. 

A glass prism of 90° is often used as a reflector. Light 

passes through the surface AB (Fig. 261) and meets the 

surface AC at an angle of 45°, which is ^ 

greater than the critical angle for glass. 

It is therefore totally reflected, and the 

device is consequently more effective y 

than an ordinary mirror. 

314. A luminous cascade. — If water 

be siphoned through a glass tube having 

an open side tubule, a (Fig. 262), and 

the tube below a be exposed to the 

direct rays of the sun or placed in the path of a beam of 

light from a lantern or porte-lumiere, the stream of water 
mingled with air which enters the 
tubule will appear like a '^stream 
of living fire," and has received the 
name of luininous cascade. This is 
due to total reflection. The light in 
passing through the water meets the 
surface of the air-bubbles at angles 
greater than the critical, and is 
reflected from side to side all down 
the stream. 

Fig. 262. 

Experiment 8. — Place uncolored glass beads, or glass broken into small 
pieces in a test-tube. They appear not only white, becaiise of diffused 
reflection, but quite opaque, because of refraction and internal reflection. 
Pour some water into the tube, and it becomes somewhat translucent. 
Substitute spirits of turpentine for the water, and the translucency is 

By mixing a small quantity of carbon bisulphide with the turpentine, 
or olive oil with oil of cassia, a liquid can be obtained whose refractive 
index is about the same as that of glass, when the light will pass through 
the liquid without obstruction, and the beads become transparent and 
nearly invisible. The last illustration shows that one transparent body 
within another can he seen only when their refractive indices differ. Place 



your eye on a level with the surface of a hot stove, and you may observe 
a w&yy motion in the air, due to the mingling of currents of heated and 

less refractive air with cooler and 
more refractive air. 

A ray of light from a heavenly 
body A (Fig. 263) undergoes a series 
of refractions as it reaches successive 
strata of the atmosphere of constantly 
increasmg density, and to an eye at 
the earth's surface appears to come 
from a point A' in the heavens. The 
general effect of the atmosphere on 
the path of light that traverses it is 
such as to increase the apparent 
altitude of the heavenly bodies. It 
enables us to see a body (B) which 
is actually below the horizon, and 
prolongs the apparent stay of the sun, moon, and other heavenly bodies, 
above the horizon, i Twilight is due both to refraction and reflection of 
light by the atmosphere. 

Fig. 2G3. 


1. Draw a straight line to represent a surface of flint glass, and draw 
another line meeting this obliquely to represent a ray of light passing 
from a vacuum into this medium. Find the direction of the ray after it 
enters the medium, employing the index as given in the above table. 

2. a. Determine the relative index of refraction for light in passing 
from water into diamond. 5. In passing from water into air. 

3. How must one modify his aim in shooting or spearing fish from the 
bank of a stream ? 

4. A ray is incident on a surface of crown glass at an angle of 40° ; 
find the angle of refraction. 

5. Find the refractive index for light passing from water into crown 

6. Does a star in the zenith appear to be where it really is ? Why ? 

1 Under average conditions the refraction elevates a body at the horizon about 35', 
so that both the sun and the moon in rising appear clear of the horizon while still 
wholly below it. The amount of refraction varies sensibly with the temperature and 
barometric pressure, increasing as the thermometer falls or as the barometer rises. 



Section VII. 


315. optical prisms. — An optical prism is a portion of a 
transparent medium bounded by plane surfaces inclined to 
each, other. Eig. 264 represents a transverse section of a 
common form of prism. Let 
AB be a ray of light incident 
upon one of its surfaces. On 
entering the prism it is re- 
fracted toward the normal, and 
takes the direction B C. On 
emerging from the prism it is 
again refracted, but now from 

the normal in the direction C D. The object that emits the ray 
will appear to be at F. Observe that the ray A B, at both refrac- 
tions, is bent toward the thicker part, or base, of the prism. 

316. Measuring iiidex of refraction. 

The ray S (Fig. 265) strikes the face A C of a prism at tlie angle 
of incidence i, and is refracted at the angle r ; sin i = n sin r, (1), n 
being the index of refraction. It strikes the face A B at the angle 

r\ and leaves the prism 
A at the angle i' \ sin i' = 

n sin r' (2). It may 
be proved geometrically 
that the angles r and r' 
are together equal to the 
angle of the prism A ; 
r+r'= A (3). Also, if 
the angle between the 
incident ray SD pro- 
duced and the deviated ray E S' be Y {i.e. the angle GO S', called the 
angle of deviation) then i +V =Y-\- A(4) . From these four equations, 
which involve n, together with any three of the six angles, z, i', r, 
r^, A, V, we may determine for any given monochromatic light 
(§ 357) the index of refraction n of the material of the prism. 

Fig. 265. 



317. Minimum deviation. 

Furthermore, it can be shown that when the prism is so placed 
that i' becomes equal to z, the angle of deviation, V, of the ray has 
its least value. Such a position is shown in Fig. 264, and is called 
the position of minimum deviation. It is easily obtained in practice 
by turning the prism until a certain position is obtained where the 
beam of light, S^ comes to a standstill and begins to move back, no 
matter which way the prism is rotated. 

318. Lenses. — Any transparent medium bounded by sur- 
faces of which, at least one is curved, is a lens. 

Lenses are of two classes, converging and diverging, ac- 
cording as they collect rays or cause them to diverge. Each 
class comprises three kinds (Fig. 266) : — 

1. Bi-convexi. 

2. Plano-convex 

3. Concavo-convex 

(or meniscus) 

Class I. 

Converging, or 
convex lenses, 
thicker in the 
middle than at 
the edges. 

Class II. 

4. Bi-concave ] Diverging, or con- 

5. Plano-concave I cave lenses, thin- 

6. Convexo-con- [ ner in the middle 

cave J than at the edges. 

A straight line normal to both surfaces of a lens and pass- 
ing through their centers of curvature, as A B, is called 

its principal axis. 
There is a point in ■ 
the principal axis 
of every lens called 
its optical cefiter. 
This point is so 
placed that a ray whose direction within the lens passes 
through it suffers no angular deviation, but at most only a 
slight lateral displacement. In lenses 1 and 4 it is half-way 
between their respective curved surfaces.^ A ray drawn 

1 If the two convex surfaces be of different curvature, the lens is called a 
■" crossed lens." 

2 In lens 2 the optical center is in its convex surface ; in lens 5 it is in its concave 
surface ; in lenses 3 and 6 it is without the lens. 

Fig. 266. 



through the optical center from any point of an object, as Ka 
(Fig. 274j, is called the secondary axis of this point. 

319. Ejfect of lenses, — We may, for convenience of illus- 
tration, regard a biconvex lens as composed, approximately, 
of a series of prisms of gradually increasing angles arranged 
around an axis, as represented in section in Fig. 267. It 
is apparent that the parallel rays 
farthest from the principal axis, /i^. 

meeting prisms of greater and /j\ 

greater angles of incidence, are 
more deflected than those nearer the 
axis ; and if the curvatures be 
properly adjusted, all may be made 
to converge to one point. 

On the other hand if the lens be 
thinnest at the center and gradually 
increase in thickness outward, exact- 
ly the opposite effect would be ex- 
pected. Parallel incident rays, being 
bent toward the thicker part of the 
component prisms, would become 

The general effect of all convex lenses is to cause transmitted 
rays to converge ; that of concave lenses, to cause them to diverge. 
Incident rays parallel to the principal axis of a convex lens 

are brought to a focus F 
(Fig. 268) at a point in the 
principal axis. This point 
is called the 'princi-pal focus, 
i.e. it is the focus of incident 
rays parallel to the principal 
axis. It may be found by holding the lens so that the rays 
of the sun may fall perpendicularly upon it, and then moving 
a sheet of paper back and forth behind it until the image of 

Fig. 267. 

Fig. 268. 



Fig. 269. 

the sun formed on the paper is brightest and smallest. Or, 
in a room, it may be found approximately by holding a lens 
at a considerable distance from a window, and regulating the 
distance so that a distinct image of the window will be pro- 

jected upon 
the opposite 
wall, as in 
Pig. 269. The 
focal length is 
the distance 
from the opti- 
cal center of 
the lens to the 
center of the 
image on the 
paper. The 

shorter the focal length the more powerful is the lens ; 
that is, the more quickly are the parallel rays that traverse 
different parts o-f the lens brought to cross one another. 

If the paper be kept at the principal focus for a short time, 
it will take fire. The reason is apparent why convex lenses are 
sometimes called 
"burning glasses." 
A pencil of rays, 
emitted from the 
principal focus F 
(Fig. 268) as a 
luminous point, be- 
comes parallel on 
emerging from a convex lens. If the rays emanate from a 
point nearer the lens, they diverge after egress, but the 
divergence is less than before ; if from a point beyond the 
principal focus, the rays are rendered convergent. A concave 
lens causes parallel incident rays to diverge as if they came 

Fig. 270. 


from a point, as F (Fig. 270). This point is therefore its 
principal focus. It is, of course, a virtual focus. 

Every lens has ^ principal focus ; it is the point to which 
parallel rays are caused to converge, or from which, after 
deflection, they appear to diverge, as the case may be. 

320. Conjugate foci. — When a luminous point beyond the 
principal focus, S (Fig. 271), sends rays to a convex lens, the 
emergent rays 
converge to an- 
other point S' ; 
rays sent from 
S' to the lens 
would converge ^^^- '^'^^• 

to S. Two points thus related are called coiyjugate foci. The 
fact that rays which emanate from one point are caused by 
convex lenses to collect at one point, gives rise to real images, 
as in the case of concave mirrors. 

321. Laiv of converging lenses. 

Lenses, like mirrors, have conjugate foci at distances p and p' 
from the optical centers. In converging lenses the principal focal 
distance and the distance of their conjugate foci (or distance of ob- 
ject and image) are related according to the formula 

V P' f 

Hence the law of converging lenses : The reciprocal of the princi- 
pal focal length is equal to the sum of the reciprocals of any two con- 
jugate focal lengths. 

When a pencil of light comes from an infinite distance (i.e. when 
its rays are parallel), p = co ; then p' =/, and the rays converge at 
the principal focus. Conversely, if a pencil come from the principal 
focus, p =f; hence _2)'= co ; that is, no image is formed. 

If the object (i.e. the source of light) be at a distance less than 
infinity, but greater than 2/, the image is real, and is on the other 
side of the lens at a distance greater than / and less than 2/ Con- 
versely, if the object be at a distance greater than /, but less than 
2/, the image is at a distance greater than 2/. 


If the object be at a distance 2/, the image is also at the distance 
. 2/, and object and image are of equal size. This suggests a simple 
way of finding /. Adjust an object, convex lens, and screen, so 
that the image on the screen is equal in size to the object. Half 
the distance of either the object or its image from the center of the 
lens is the focal length of the lens. 

322. Diverging lenses. 

The formula for these is "> = 7 ' 

P V f 

When p = CO , p' = — / (a virtual image at the principal focus i). 
When _p = /, _p' = — CO (no image). 

When p is of any value greater than /, and less than co , _p' Is 
greater than /. 

323. Images fanned. — Fairly distinct images of objects 
may be formed through very small apertures (§ 283); but 
owing to the small amount of light that passes through the 
aperture, the images are very deficient in brilliancy. If the 
aperture be enlarged, brilliancy is increased at the expense of 
distinctness. A convex lens enables us to obtain both brilliancy 
and distinctness at the same time. 

Experiment 1. — By means of a porte-lumi^re, A (Fig. 272), introduce 
a horizontal beam of light into a darkened room. In its path place some 
object, as B, painted in transparent colors or photographed on glass. 
(Transparent pictures are cheaply prepared by photographers for sunlight 
and lime-light projections.) Beyond the object place a convex lens L, 
and beyond the lens a screen S. The object being illuminated by the 
beam of light, all the rays diverging from any point a are bent by the 
lens so as to come together at the point a'. In like manner, all the rays 
proceeding from c are brought to the same point c' ; and so also for all 
intermediate points. Thus, out of the billions of rays emanating from 
each of the millions of points on the object, those that reach the lens are 
guided by it, each to its own appropriate point in the image. It is 
evident that there must result an image both bright and distinct, pro- 
vided the screen be suitably placed, i.e. at the place where the rays meet. 

1 The negative sign refers to the direction in which p' is measured. Conjugate 
foci of diverging lenses are on the same side of the lens. 



But if the screen be placed at S' or S", it is evident that a blurred image 
will be formed. Instead of moving the screen back and forth, in order 
to " focus " the rays properly, it is customary to move the lens. 

Experiment 2. — Make a series of experiments similar to those with 
the concave mirror. Ascertain the focal length of the convex lens. 


Fig. 272. 

Place the lens at a distance from a white wall about equal to its focal 
length. Place a candle flame (better the flame of a fish-tail burner) at 
such a distance the other side of the lens that it will produce a distinct 
and well-defined image on the wall (Fig. 273). (1) Observe and note on 


Fig. 273. 

paper the size and kind of image. Advance the flame toward the lens, 
regulating at the same time the distance between the lens and wall, so as 
to preserve a distinctness of image. (2) Note the changes which the 



image undergoes. (3) When the image and the flame become of the 
same size, measure and note the distance of each from the lens. (4) Ad- 
vance the flame still nearer, and note the changes in the image, until it is 
impossible to obtain an image on the wall. Measure the distance of the 
flame from the lens, and compare this distance with the focal length of 
the lens. (5) Move the flame still nearer. Note whether the rays, after 
emerging from the lens, are divergent or convergent. (6) See whether an 
image and an object may change places. (7) Form images of the flame 
on the wall at different distances from the lens ; measure the distances, 
also the linear dimensions {e.g. the width, or the vertical hight) of the 
images, and determine whether the linear dimensions of images are pro- 
portional to their distances from the lens. 

324. To construct the image formed by a convex lens. — 
Given the lens L (Fig. 274), whose principal focus is at E, 
and object AB in front of it ; any two of the many rays from 

Fig. 274. 

A will determine where its image a is formed. Two that can 
be traced easily are, one along the secondary axis A a, and 
one parallel to the principal axis A A': the latter will be 
deviated so as to pass through the principal focus F, and 
will afterward intersect the secondary axis at some point a ; 
therefore this is the conjugate focus of A. Rays can be 
similarly traced for B, and all intermediate points along the 
arrow. Thus, a real inverted image is formed at a h. 

The linear dimensions of an object and of its image formed 
by a convex lens are jproportional to their resjicctive distances 
from the center of the lens. The image is virtual or real, erect 
or inverted, according as it is on the same side of the lens 
with the object or on the op^josite side. 


325. Virtual images. — Since rays that emanate from a 
point nearer the lens than the principal focus diverge after 
egress, it is evident that their focus must be virtual and on 
the same side of the lens as the object. Hence, the image of 
an object placed nearer the lens than the principal focus is 
virtual^ magnified, and erect, as shown in Fig. 275. A convex 
lens used in this manner is called a simple microscope. 

326. Simple microscope. As its name implies, the micro- 
scope is an instrument for viewing minute objects. The 
simple microscope consists of a single converging lens so 
placed that the object is between the principal focus and the 
lens. It magnifies by increasing the visual angle. 


A ,-'-' 

-,B ^---^ 

Fig. 275. 

The magnifying poiuer of the lens is simply the ratio 
between the apparent diameter of the image and the diameter 
of the object, e.g. A'B' : AB (Fig. 275), or it is the ratio 
between the visual angles under which the eye would see 
image and object, if both were placed at the distance of 
distinct vision.^ If the lens be of short focus, as is usually 
the case, the magnifying power is approximately the ratio of 
the distance of distinct vision to the focal length. Thus a 
lens of -J- in. focal length would magnify 20 to 24 diameters. 

- 1 For normal eyes, an object to be seen most distinctly must be placed at a 
distance of 10 to 12 inclies, bence this is regarded as the distance of distinct vision. 



327. Diverging lenses. — Since the effect of concave lenses 
is to render transmitted rays divergent, pencils of rays 
emitted from A and B (Fig. 276) diverge after refraction, as 
if they came from A' and B', and the image will appear to be 
at A' B'. Hence, images formed hy concave lenses are virtual, 
erect, and smaller than the object. 

Fig. 276. 

328. Spherical aberration. — In all ordinary convex lenses 
the curved surfaces are spherical, and the angles which inci- 
dent rays make with the little plane surfaces of which we 
may imagine the spherical surface to be made up, increase 

Fig. 277. 

rapidly toward the edge of the lens. Thus, while those rays 
from a given point of an object which pass through the cen- 
tral portion, as A (Fig. 277), meet approximately at the same 
point F, those which pass through the marginal portion are 
deflected so much that they cross the axis at nearer points, 
e.g. at F' ; so a blurred image results. This wandering of the 
rays from a single focus is called spherical aberration. 

No lens with spherical surfaces can bring all the rays to 
the same focus. The may be in a measure corrected by 


interposing a diaphragm D D' provided with a central aperture 
smaller than the lens, so as to cut off those rays that pass 
through the marginal part of the lens. But it can be wholly 
corrected only by properly modifying the curvature of the 
surfaces of the lens. A lens having surfaces thus modified is 
said to be aplanatic. 

Experiment 3. — (Illustrating spherical aberration.) Cut a cardboard 
disk as large as the convex lens to be employed. Cut a ring of holes 
near the circumference, and also a ring near the center. Support the 
disk close to the lens, so as to cover one of its surfaces. Place the whole 
in a beam from a porte-lumiere. Catch refracted beams on a screen. 
Move the screen away from the lens. The beams through the outer ring 
of spots are the first to cross one another and form an image. Further 
away, the inner beams coincide, forming an image. The outer ones, 
having crossed, form a ring of spots. 


1. What must be the position of an object with reference to aeon- 
verging lens, that its image may be real and magnified ? 

2. A photographic transparency is placed between a porte-lumi6re and 
a biconvex lens, 16 in. from the latter ; how many diameters is a distinct 
image of the transparency multiplied on a screen 20 ft. distant ? 

3. A transparency whose dimensions are 3 X 4 in. is placed 16 in. from 
the lens ; at what distance from the lens must the screen be that it may 
receive a distinct image of the transparency that shall cover a surface 
3 X 4 ft. ? 

4. What is the focal length of the lens used in the last case ? 

5. With a converging lens the image of a candle is thrown on a screen 
6 ft. from the candle, and the focal length of the lens is 16 in. ; find the 
distances of the candle and of the screen from the lens. Ans. 4 ft. and 2 ft. 

6. A luminous point is 3 in. from a convex lens having a focal length 
of 5 in. ; find the position of the image. 

7. If the candle and screen be 3 ft. apart, and the lens midway between 
them, what is the focal length ? 

8. Find the focal length of a lens which throws the image of an object 
5 ft. distant on a screen 3 ft. distant. 

9. A double concave lens having a focal length of 3 in. is held at a 
distance of 2 in. from a small object ; find the position of the image. 



10. If an object be at twice tlie focal distance of a convex lens, how 
will the length of the image compare with the length of the object ? 

11. To an eye whose distance of distinct vision is 25 cm, how many- 
diameters will a lens of 1 cm focus magnify ? 

12. Show that a concave air lens in water has the same effect on inci- 
dent light as a convex water lens in air. 

Section VIII. 


329. Analysis of light which produces the sensation of white. 

JExperiment 1. — Place a disk with an adjustable slit in the aperture of 
a porte-lumiere, so as to exclude from a darkened room all light-waves 
except those which pass through the slit. Near the slit interpose a 

Fig. 278. 

double-convex lens of (say) 10-inch focus. A narrow sheet of light will 
traverse the room and produce an image, AB (Fig. 278), of the slit on a 
white screen placed in its path. Now place a glass prism C in the path 
of the narrow sheet of light and near to the lens, with its edge vertical. 


(1) The light now is not only turned from its former path, but that which 
before was a narrow sheet, is, after emerging from the prism, spread out 
fan-like into a wedge-shaped body, with its thickest part resting on the 
screen. (2) The image, before only a narrow, vertical band, A B, is now 
drawn out into a long horizontal ribbon DE. (3) The image, before 
white, now presents all the colors of the rainbow, from red at one end 
to violet at the other ; it passes gradually through all the gradations of 
orange, yellow, green, blue, and violet. (The difference in deviation 
between the red and the violet is purposely much exaggerated in the 

From this experiment Ave learn (1) that ivhite light is not 
simple in its composition, hut the result of a mixture of colors} 

(2) The colors of luhich white light is composed may be sepa- 
rated hy refraction. (3) The separation is due to the different 
degrees of deviation which colors undergo hy refraction. Eecl, 
which is always least turned aside from a straight path, is 
the least refrangible color. Then follow orange, yellow, 
green, blue, and violet, in the order of their refrangibility. 
The many-colored ribbon of light DE is called the solar spec- 
trum? This separation of white light into its constituents is 
called dispersion. The number of colors of which white light 
is composed is really infinite, but we have names for only 
seven of them ; viz. red^ orange^ yellow, green, cyan-hlue, ultra- 
marine-hlue, and violet ; and these are called the primary or 
prismatic colors. The names of the blues are derived from 
the names of the pigments which most closely resemble them. 

The spectrum may be projected upon a screen, or it may be 
received directly by the eye, as in the two following experi- 
ments : — 

Experiment 2. — Upon a black cardboard A (Fig. 279) paste a strip 
of white paper 3 cm long and 2 mm wide ; and place the prism and the 
eye as in the figure. Now when a beam of white light from the strip is 

1 Newton (1666) was the first to recognize the true import of this phenomenon, i.e. 
to refer the colors to the heterogeneity of white light. 

2 A succession of colors in the order of their refrangibility, obtained from any 
source of light, is called a spectrin. 



refracted and dispersed by the prism and falls upon the retina of the eye, 
you see, not the narrow white strip in its true position, but a spectrum 
in the position A^ This experiment is performed in a lighted room. 

Experiment 3. — Instead of a continuous white 
strip, paste short strips of red, white, and blue, 
end to end, on the black card, as represented in 
Fig. 280. The spectrum of each color is given on 
the right, the light portions representing the illu- 
minated parts. It will be seen 
that in the spectrum of the red, 
the green, blue, and violet 
portions are almost completely 
dark, but there is a faint trace 
of orange ; in the spectrum of 
the blue, the red, orange, and 
yellow are wanting, blue and 
violet are present, and a small 
quantity of green. 






Fig. 279. 

Fig. 2g 

330. Synthesis of white light. — The composition of white 
light has been ascertained by the process of analysis ; it can 
be verified by synthesis ; i.e. the colors after dispersion may 
be reunited, and the result of the reunion is white light. 

Experiment 4. — Place a second prism (2) in such a position /^ that 
light which has passed through one prism (1), and been refracted and 
decomposed, may be refracted back, and the colors will be reblended, 
and a white image of the slit will be restored on the screen. 

Experiment 5. — Place a large convex lens, or a concave mirror 
(better a concave cylindrical mirror), so as to receive the colors after dis- 
persion by a prism, and bring the rays to a focus on a screen. The 
image produced will be white. 

Experiment 6. — Receive the spectrum on a common plane mirror, 
and rapidly tip the mirror back and forth in small arcs, and the light 
reflected by the mirror upon a screen will produce a white image on the 

331. The rainbow. — The rainbow is a solar spectrum on a 
grand scale. It is the result of refraction, total reflection, 



and dispersion, of sunlight by falling raindrops. Let spheres 
1 and 2 (Fig. 281) represent drops at the extreme opposite 
edges of the bow. The eye is in a position to receive, after 
the dispersion and internal reflection of the light-waves within 
drop 1, only the red waves ; consequently this part of the 
bow appears red. So, likewise, from drop 2 the eye receives 
only violet ; consequently this part of the bow appears violet. 
In like manner, the intermediate colors of the bow are 
sifted out. 

Outside the primary bow a secondary/ bow is sometimes seen. 
Drops 3 and 4 (Fig. 281) are supposed to be at the opposite 

Fig. 281. 

edges of the secondary bow. It will be seen that the light- 
waves undergo two internal reflections within the drops 
which produce this bow. The colors of this bow are in 
reverse order to those of the primary bow, and less brilliant. 
332. Chromatio aberration. — There is also in ordinary 
convex lenses a serious defect, to which we have not before 
referred, called chromatic aberration, the correction of which 
has demanded the highest skill. The convex lens both refracts 


and disperses the light-waves that pass through it. The ten- 
dency, of course, is to bring to a focus the more refrangible 
rays, as the violet, much sooner than the less refrangible 
rays, such as the red. The result is a disagreeable coloration 
of the images that are formed by the lens, especially by those 
portions of the light-waves that pass through the lens near 
-.^^^^^^^ ^^^^^^^ its edges. This evil has been overcome very 
^ .^-^r^r-r-^-,^^^ effectually by combining with the convex lens 
a plano-concave lens. Now, if a crown-glass 
jjiG. 282. convex lens be taken, a flint-glass concave 

lens may be prepared that will correct the dispersion of the 
former without neutralizing all its refraction.^ A compound 
lens composed of these two lenses cemented together (Fig. 282) 
constitutes what is called an achromatic lens. 

333. Cause of color and dis2)ersio7i. — The color of light is 
determined by vibration-frequency, or, in other words, by the 
corresponding wave-length. The light-waves diminish in 
length from the red to the violet. As pitch depends on 
the frequency with which aerial waves strike the ear, so color 
depends upon the frequency with which ether-waves strike 
the eye. The difference between violet and red is a difference 
analogous to the difference between a high note and a low 
note on a piano. 

The speed of propagation in a vacuum appears to be the 
same for all wave-lengths. But in a refracting medium, the 
short waves are more retarded than the longer ones, hence 
they are more refracted. This is the cause of dispersion. 
Each wave-length has its own refractive index, or, since 
vibration-frequency corresponds to color, every simple color 
has its special refractive index. Light composed of waves all 
of the same (or nearly the same) length is called homogeneous 
or monochromatic light. The yellow light emitted by the 
flame of a Bunsen burner or alcohol lamp when common salt 

1 The refractive and dispersive powers of the two lenses are not proportional. 


is sifted upon it is approximately monochromatic. Ordinary 
white light is a mixture of long and short ether-waves. 

From well-established data, determined by a variety of 
methods, physicists have calculated the number of waves that 
succeed one another for each of the several prismatic colors, 
and the corresponding wave-lengths ; the following table con- 
tains the results.^ The letters A, C, J), etc. refer to Fraun- 
hofer's lines (see § 340). 

Length of waves No. of waves 

in millimeters. per second. 

Dark red A 000760 395,000,000,000,000 

Orange C .000656 458,000,000,000,000 

Yellow D 000589 510,000,000,000,000 

Green E 000527 570,000,000,000,000 

C. Blue F 00048G 618,000,000,000,000 

U. Blue G 000431 697,000,000,000,000 

Violet H 000397 760,000,000,000,000 

There is a limit to the sensibility of the eye as well as of 
the ear. The limit in the number of vibrations appreciable 
by the eye lies approximately within the range of numbers 
given in the above table ; i.e. if the succession of waves be 
much more or much less rapid than is indicated by these 
numbers, the sensation of sight is not produced. 

" Our knowledge of ether-waves is at present limited to 
those which lie between 107 trillions and 40,000 trillions per 
second — a range, in musical parlance, of about 8^ octaves " 
(Langley). Of these our eyes are sensitive to scarcely one 

It is evident that the frequency of the tvaves emitted hy a 
luminous body, and consequently the color of the light emitted, 

1 " That man should he able to measure, with certainty, such minute portions of 
space and time, is not a little wonderful ; for, whatever theory of light we adopt, it 
may he observed that these periods and these spaces have a real existence, being, in 
fact, deduced by Newton himself from direct measurements, and involving nothing 
hypothetical but the names here given them." — Sir John Herschel. 

If science in the future shall be able to dispense with the ether of space, the 
vibration periods and what corresponds to wave-lengths will necessarily remain. 


must depend on the rapidity of the vibratory jnotions of the 
molecules of that body, i.e. upon its temperature. This has 
been shown in a convincing manner* as follows : The temper- 
ature of a platinum wire is slowly raised by passing a 
gradually increasing current of electricity through it. At a 
temperature of about 540° C. it begins to emit light ; and if 
the light be analyzed by a prism, it is shown that only red 
light is emitted. As the temperature rises, there will be 
added to the red of the spectrum, first yellow, then green, 
blue, and violet successively. When it reaches a white heat, 
it emits all the prismatic colors. It is significant that a 
white-hot body emits more red light than a red-hot body, and 
likewise more light of every color than at any lower temper- 
ature. The conclusion is, that a body which emits luhite light 
sends forth simultaneously tuaves of a variety of lengths. 

334. Continuous spectrums. — The spectrum produced by 
the platinum is continuous ; that is, the band of light is 
unbroken. If the spectrum be not complete, as when the 
temperature is too low, it will begin with red, and be con- 
tinuous as far as it goes. All luminous solids and liquids give 
continuous spectrums. 

A gas, kerosene, or candle flame does not give the spectrum 
of a vapor, but gives that of the solid particles of carbon in a 
state of incandescence ; hence the continuous spectrums which 
these flames afford. 

335. Spectroscopes. — Instruments for the observation of 
spectrums are called spectroscopes. The essential part of the 
apparatus is the "dispersion piece," which is either a prism 
or a diffraction grating (see Fig. 302). Instead of looking at 
the spectrum with the naked eye it is usually better to view 
it through a small telescope, which serves to magnify it. 
Fig. 283 represents the simplest form of the Kirchhoff and 
Bunsen spectroscope. A flint glass prism receives light 
through an adjustable slit at the end of a tube called the 


collimator. At the opposite end of this tube is a converging 
lens, and the slit is located at its principal focus so that rays 
diverging from the slit are rendered parallel by the lens. 

It is often necessary to have some means of determining 
the positions of certain lines (to be described hereafter) 

Fig. 283. 

observed in the spectrum. The usual method is to have a 
second tube, somewhat like the collimating tube, so placed 
that the rays from a light (e.g. a candle flame as in Fig. 
283) after passing through a transparent plate (inside the 
tube) on which a fine scale is engraved, and through a lens, 
by which they are made parallel, are reflected from the 



nearest face of the prism, and pass into the telescope along 
with the beam of light under analysis. Thus the eye while 
viewing the spectrum through the telescope sees also a 
magnified image of the scale coinciding with the spectrum. 

336. Direct -vision pocket spectroscope. — A small instru- 
ment called a pocket spectroscope will answer fairly well for 
experiments given in this book. This instrument contains 
three or more prisms, A, B, and C (Fig. 284). The prisms 
are enclosed in a brass tube, D, and this tube in another tube, 
E. F is a convex lens, and G is an adjustable slit. By 
moving the inner tube back and forth, the instrument may 

Fig. 284. 

be so focused that parallel rays will fall upon prism A. This 
instrument has no telescope. By varying the kind of glass 
used in the different prisms,^ as well as their structure, the 
deviation of light from a straight path in passing through 
them is overcome, while the dispersion is preserved. On 
account of the directness of the path of light through it, this 
instrument is called a direct-vision spectroscope. 
337. Bright line spectrimis. 

Experiment 7. — Open the slit about one-sixteentli of an inch wide, 
by turning the milled ring M (Fig. 285), and look through the spectro- 
scope at the sky (not at the sun, for its light- 
waves are too intense for the eye); you will 
see the solar spectrum. 

Experiment 8. — Repeat the last experiment 
with a candle, kerosene, or ordinary gas-flame, 

Fig. 285. 

and you will obtain similar results. 

1 A and C are crown-glass, and B is flint-glass. See footnote, p. 372. 


Experiment 9. — Take a piece of platinum wire 2 inches long. Seal 
one end by fusion to a short glass tube for a handle. Bend the wire at 
a right angle. Dip a portion of the wire into a strong 
solution of common salt, and support it by a clamp in a 
the midst of the almost invisible and colorless flame of (iliL.^^^^ 
a Bunsen burner (Fig. 286). Instantly the flame ^11 ^""""^^^ 
becomes luminous and colored a deep yellow. Examine | 
it with a spectroscope, and you will find, instead of a I 
continuous spectrum beginning with red, only a bright, M, ^^^^^ 
narrow line of yellow, in the yellow part of the spectrum, ^^J^^^^l 
next the orange. Your spectrum consists essentially of ^^^^''~ 
a single! bright yellow line on a comparatively dark Fig. 286. 

ground (see Sodium, Plate I, frontispiece). 

Experiment 10, — Heat the platinum wire until it ceases to color the 
flame, then dip it into a solution of chloride of lithium, and repeat the last 
experiment. You obtain a carmine-tinted flame, and see through the 
spectroscope a bright red line and a faint orange line (see Lithium, Plate I). 

Experiment 11. — Use potassium hydrate, and you obtain a violet- 
colored flame, and a spectrum consisting of a red line and a violet line 
(the latter very difiicult to see even with the best instruments). Use 
strontium nitrate, and obtain a crimson flame, and a spectrum consist- 
ing of several lines in the red and the orange and a blue line (see Potas- 
sium and Strontium, Plate I). 

Experiment 12. — Use a mixture of several of the above chemicals, 
and you will obtain a spectrum containing all the lines that characterize 
the several substances. 

Every chemical compound used in the above experiments 
contains a different metal, e.g. common salt contains the 
metal sodium ; the other substances used successively con- 
tain respectively the metals lithium, potassium, and stron- 
tium. These metals, when introduced into the flame, are 
vaporized, and we get their spectrums when in a gaseous 
state. All incandescent gases, unless %inder great 'pressure, 
give discontinuous, or bright line, sjpectrunfis, and no two gases 
give the same spectrum. 

1 Spectroscopes of higher dispersive power show that the sodium line is really a 
double line divided by a narrow interval. 

" It is not a hypothesis, but a reality, that sodium vapor has two independent 
vibrations, whose periods differ by about ^q^otj of each other." —Lord Kelvin. 


338. Spectmm analysis. 

A vibrating molecule embedded in ether emits waves, the length 
of which depends on the rate of vibration, and the waves of different 
lengths produce the different color sensations. Now, like a tuning 
fork, a free molecule of every substance has its own definite period 
or periods of vibration, and accordingly sends out light of a certam 
definite color or of a few definite colors, just as the fork emits sound 
of a certain definite pitch, with sometimes a few harmonics. For 
example, every molecule of sodium or of lithium vibrates in the 
same way, and always has vibrated in the same way, whether it 
exists in the sun, in the earth, or in a distant star. The same is 
true of every other kind of matter ; each has its own rates of vibra- 
tion, and hence each produces its own bright line spectrum corre- 
sponding with its peculiar rates of vibration. Hence has arisen a 
new chemical analysis, wherein substances are detected simply by 
observing the rates of vibration of their molecules {i.e. the bright 
lines of their spectrums), a branch of physical chemistry called 
spectrum analysis. 

It is only in the gaseous state, however, that the molecule is free 
to exhibit its special rate of vibration ; when they are packed closely 
together in a solid or liquid, their motions are cramped, their perio- 
dicity is lost, and all manner of vibrations are induced. Hence 
spectrums of solids and liquids are continuous, i.e. the rates of 
vibrations are so many in number as to leave no gaps in their 

Many chemical compounds are decomposed into their elements, 
and the elements are rendered gaseous at a temperature that is at, 
or below, the temperature necessary for incandescence. In that 
case the spectrum given is the combined spectrums of the elements. 
A compound gas that does not suffer dissociation at the temperature 
of incandescence gives its own spectrum, which is generally totally 
different from the spectrum of its elements. 

339. Reversed or dark line spectrum. 

Experiment 13. — Arrange apparatus in a dark room as in Fig. 287. 
N is the flange nozzle of a stereopticon (p. 432) containing only the con- 
densing lens ; T and S are two tin plates, in the latter of which a slit is 
cut. Allow a beam of calcium light to pass through the slit in S, and 
thence through the converging lens L and the prism P, and form a spec- 
trum on a screen, H. Hold in the flame of a Bunsen burner, B, a pellet 



of sodium ; it burns vividly, and the light has to pass through the in- 
tensely yellow flame. We should naturally expect that the yellow of the 
spectrum would now be more intensely illuminated, but, instead, a dark 
band in the yellow now appears. It is not really black, but compara- 
tively dark. 

Next hold the plate T between the burner and the condensers so that 
the calcium light may be cut off from the upper portion of the slit, 
leaving the light from the sodium flame alone to pass through this part 
of the slit, The spectrum R formed by this part consists of a bright 
yellow line on a dark ground, being the radiation spectrum of sodium. 
(It should be borne in mind that the image of the slit is inverted.) The 

Fig. 287. 

other half, A, shows a dark line on the continuous spectrum. We thus 
have, contiguous to each other, the bright line spectrum of sodium and 
its reversed^ dark line, or absorption spectrum. If you use salts of lithium, 
potassium, strontium, etc., in a similar manner, you will find in every 
case your spectrum crossed by dark lines where you would expect to find 
bright lines. 

It thus appears that the vapors of different substances absorb 
or quench the very same rays that they are capable of emitting 
when made self-luminous; very much, it would seem, as a 
given tuning-fork selects from various sounds only those of 


a definite wave-length corresponding to its own vibration- 
period. The dark places of the spectrum receive light in 
full force from the salted flame.; but the light is so feeble, 
in comparison with those places illuminated by the calcium 
light, that the former appear dark by contrast. Light trans- 
mitted through certain liquids (as sulphate of quinine and 
blood) and certain solids (as some colored glasses) produces 
band spect-rums. These spectrums are obtained only when 
light passes through mediums capable of absorbing rays of 
certain wave-length ; hence, they are commonly called absorp- 
tion spectrums. Since a given vapor causes dark lines pre- 
cisely where it would cause bright lines if it were itself the 
only radiator of light, dark line spectrums are frequently 
called reversed spectrums. There are then three kinds of 
spectrums : continuous spectrums, produced by luminous solids, 
liquids, or, as has been found in a few instances, gases under 
great pressure ; bright line spectrums, produced by luminous 
vapors ; and absorption spectrums, produced by light that has 
been sifted by certain mediums. 

340. Fraunhofer^s lines. — The spectrum of sunlight is 
observed to contain a large number of dark lines transverse 
to its length. These were first observed by Wollaston (1802), 
and were mapped by Fraunhofer (1814) who distinguished 
several of the more prominent ones by letters of the alpha- 
bet ; hence the dark lines of the solar spectrum have received 
the Dame of Fraunhofer' s lines. 

So far as discovered, no two substances have a spectrum 
consisting of the same combination of lines ; and, in general, 
different substances very rarely possess lines appearing to 
be common to both. Hence, when we have once observed and 
mapped the spectrum of any substance, we may ever after be 
able to recognize the presence of that substance when emit- 
ting light, whether it is in our laboratory or in a distant 
heavenly body. 

fkaunhofer's lines. 381 

The spectroscope, therefore, furnishes us a most efficient 
means of detecting the presence (or absence) of any elemen- 
tary substance, even when it is combined or mixed with other 
substances. It is not necessary that the given substance 
should exist in large quantities ; for example, the fourteen- 
millionth part of a milligram of sodium can be detected by 
the spectroscope. Substances which are not easily converted 
into vapors at low temperatures may be placed between the 
poles of an electric battery or an induction coil. The heat 
generated by electricity will vaporize most substances. Thus 
the spark passing between two copper electrodes will vaporize 
a portion and show the copper lines, between iron electrodes 
the iron lines, etc. After maps of the spectrum of all known 
substances have been made out, if, on examination of a com- 
plex substance, any new lines should at any time appear in 
the spectrum, it would indicate the presence of a substance 
hitherto undiscovered. It was thus that the elements cae- 
sium, rubidium, thallium, and indium were discovered.^ 

341. General remarks. 

Gases or vapors when sufficiently heated to become luminous 
emit, under ordinary pressure, color rays which are dispersed into 
an interrupted spectrum of bright lines ; with increasing pressure 
and density these lines, spread into diffuse luminous bands, and 
finally form a continuous spectrum. Gases are rendered luminous 
usually by passing electric sparks through glass tubes enclosing 
them. Substances whose great volatility interferes by causing 
evaporation before the substpvnce attains the temperature of incan- 

1 Fraunliof er's lines, designated by letters of the alphabet, beginning at the red 
end of the spectrum, have the follo^ving wave-lengths, expressed in million ths of a 
millimeter : 

Line. Wave-length. Line. Wave-length. 

A 762.1 Ej 527.0 

B 688.4 b, 518.3 

C . 656.3 F 486.1 

Di 589.6 G 430.7 

Da 589.0 H, 396.8 

H, 393.3 


descence, such as most metalloids, give no flame emission spectrums. 
These are commonly rendered luminous in glass tubes called Plticker 
tubes (Fig. 288). 

Colored liquids are usually placed in glass cells with flat parallel 
sides, or of wedge shape so as to allow the examination of different 
thicknesses of the liquid. The cells are placed between —>. 
a bright flame and the slit (or better in reflected sunlight), A 
and the result is absorption spectrums of the coloring ( j 
matter, consisting of dark bands. Thus blood in its 
normal or healthy state is readily recognized by the 
absorption spectrums of its coloring matters and their 
modifications by absorption of gases, i Aniline blue 
shows a very dark absorption band from wave-length 
656 (C) to 550, gradually becoming lighter from there to 
520 (just beyond b'). 

The spectroscope is very useful to the pathologist in 

examining diseased blood, in detecting albumen in urine, — ' 

Fig 288 
in investigating supposed cases of poisoning, etc. ; to the 

merchant in distinguishing certain liquids such as wine, beer, etc. 

in the normal state and the adulterated state. 

342. Solar and stellar chemistry and plujsics. 

The spectrum of iron has been mapped to the extent of more 
than 600 bright lines. Of these, Kirchhoff succeeded in showing 
the coincidence of 460 with dark lines of the solar spectrum. Can 
there be any doubt of the existence of iron in the sun ? By exam- 
ination of the reversed spectrum of the sun, we are able to deter- 
mine with certainty the existence there of sodium, calcium, copper, 
zinc, magnesium, hydrogen, and many other known substances. 
Again, from our knowledge of the way in which a reversed spectrum 
can be produced, we may conclude that the sun consists of a lumi- 
nous solid, a liquid, or an intensely heated and greatly condensed 
gas (called a p/io^osp/zere), and that this nucleus is surrounded by an 
atmosphere of cooler vapor, in which exist at least all the substances 

lA solution of fresh blood gives two easily distinguishable dark bands in the 
green. But blood is capable of existing in different stages of oxidation which are 
distinguishable by difference of color and corresponding difference of spectrum. By 
means of a microspectroscope (a combination of microscope and spectroscope) it is 
claimed that the thoiisandth part of a grain of blood is easy of detection, and its pres- 
ence may be detected in stains that have been kept a very long time ; hence this in- 
strument often becomes of great importance in criminal trials. 


just named. The moon and planets that are visible only by reflected 
sun-light give the same spectrums as the sun, while those that are 
self-luminous give spectrums which differ from the solar spectrum. 

343. Effect of ^notion in the radiating hody. 

If the radiating body be in motion, either to or from the observer, 
obviously the effect of this motion will be to shorten the wave-length 
in the former case and to lengthen it in the latter (compare §173). 
This will tend to move the spectrum lines toward the more re- 
frangible end in the first instance and toward the less refrangible 
end in the latter. Thus from the displacement of hydrogen lines in 
the absorption spectrum Young computes that the greatest speed 
upon the sun observed by him is 400 kilometers per second. 
Pickering (Harvard Univ.) has observed that the Tc line in the spec- 
trum of j8 Aurigae is alternately single and double at intervals of 
about seventeen hours, thus showing this star to be double, each 
part revolving about the other in less than four days at a speed 
of 240 kilometers per second. 

The telespedroscope (a combination of telescope and spectroscope) 
has disclosed to us a much-coveted knowledge of the true nature, 
chemical composition, and physical condition of those points of 
light called the fixed stars, immensely more remote and less bright 
than the planets. In like manner nebulae, comets, and meteors 
have been investigated, and valuable knowledge has been obtained 
as to their physical constitution. 

344. Distribution of energy in the spectrum. — The energy 
of ether waves is capable, as has been before observed, of pro- 
ducing calorific, luminous, or chemical effects, according to 
the nature of the bodies upon which it falls. When a sensi- 
tive thermoscope is passed along the spectrum, heat effects 
are observed throughout the visible spectrum, and for con- 
siderable distances beyond at each extremity. All ether 
waves are capable of producing heating effects. 

It thus appears that the solar spectrum is not limited to the 
visible spectrum, but extends beyond at each extremity, and 
spectroscopic analysis, besides sifting the waves of one color 
from those of another, is able to sift out rays which do not 


produce the sensation of light from those which do. Those 
rays that lie beyond the red are called the infra-red rays, while 
those that lie beyond the violet are called the ultra-violet rays. 
The infra-red rays are of longer vibration period, and the 
ultra-violet of shorter period, than the luminous waves. 

Inasmucli as glass largely absorbs the energy of ether waves of 
certain lengths, it is customary in studying heat spectrums to use 
lenses and prisms of rock saft, since this substance transmits waves 
of all lengths with great freedom. 

In a prismatic spectrum obtained by the use of a rock salt prism, 
the maximum heating effect for the solar spectrum is in the infra- 
red. Langley finds, however, that in the normal spectrum the maxi- 
mum heating effect in the solar radiation coincides quite closely 
with the maximum luminous effect which is in the orange-yellow. 

Chemical effects are produced by rays of all refrangibilities. 
A photograph can be taken of all portions of the visible spectrum, 
and the photographic spectrum may extend far beyond the visible 
spectrum in the ultra-violet, and even the infra-red rays may be 
photographed. Ordinary silver salts are decomposed by rays ex- 
tending from the green upward, while the decomposition of carbon 
dioxide is aided chiefly by rays of lower refrangibility. 

345. Only one kind of radiation. ■ — The fact that radiant 
energy produces three distinct effects, — viz. luminous, heat- 
ing, and chemical, — has given rise to a prevalent idea that 
there are three distinct kinds of radiation.^ There is, how- 
ever, absolutely no proof that these different effects are pro- 
duced by different kinds of radiation. Science recognizes in 
radiations no distinctions but periods, wave lengths, and wave 
forms. The same radiation that 2>^'oduces vision can generate 
heat and chemical action. The fact that the infra-red and ultra- 
violet rays do not affect the eye does not argue that they are of 
a different nature from those that do, but it does show that 

1 One great service which the diifraction spectrum (see § 361) has rendered to science 
is the aholishment of all these imaginary independent existencies — heat, light, acti- 
nism, etc., and the suhstitiition for them of the far simpler conception of vibratory 
motions of ether differing only in rate of vibration, the diversity of effects produced 
depending on the quality of the surface on which they fall. 


there is a limit to the susceptibility of the eye to receive im- 
pressions from radiation. Jnst as there are sound-waves of 
too long, and others of too short period to affect the ear, so 
there are ethereal waves, some of too long, and others of too 
short period to affect the eye. 

346. PJiosphorescence. — There is a class of substances such 
as the sulphides of calcium, strontium, etc., which after 
several hours' exposure to light-waves absorb their energy 
(i.e. their molecules acquire sympathetic vibrations) without 
becoming hot, and in return emit light-waves, which are quite 
perceptible in a dark room for several hours after the ex- 
posure. This property of shining in the dark after having 
been exposed to light-waves is termed pJiosphorescence. A so- 
called Imninous jpahit is prepared and applied to certain parts 
of bodies that are exposed to sunshine during the day ; at 
night those parts to which the paint is applied are alone 
luminous. This paint may be used for a variety of purposes, 
such as rendering luminous danger signals, door numbers and 
plates, etc. 

347. Fluorescence. 

There is another class of substances which are acted upon in a 
somewhat similar and yet somewhat different manner. The vibra- 
tions of the extremely small ether-atoms may set up in the more pon- 
derous molecules of matter slower (forced) vibrations. We have an 
example of this when light-waves are absorbed by a body and it in 
turn emits only the longer, invisible waves. Much as short, choppy 
waves acting upon a vessel anchored at sea impart a slower pitching 
and rolling of the vessel, and these in turn new and slower waves in 
the water, the invisible ultra-violet waves may give rise to dis- 
turbances of the interior molecules of a body impinged upon, and 
give rise to other waves which are not so frequent as to be invisible. 
In this case the body emits light from within, and in some cases 
continues to emit light for some seconds after the light is shut off. 
This phenomenon is known by the name fluorescence, from fluor- 
spar, one of the first substances in which it was observed. Among 
fluorescent substances are aesculin (derived from the bark of the 


horse-chestnut), quinine sulphate, sulphides of barium and calcium, 
diamond, uranium glass, etc. These substances when illuminated 
by certain rays of the spectrum in a darkened room shine from 
within with a lustre of their own, each showing its own special 
color, a color not by any means the same as the natural color of 
the body itself as seen in the open white light of day. This is best 
accomplished by means of the light of electric discharges. For this 
purpose solids are usually enclosed in the so-called Geissler tubes 
and an electric current is passed through them. 

348. Calorescence. 

On the other hand, slow waves may be transformed into more 
rapid ones. A beam of light passing through a solution of iodine 
in carbon bisulphide loses all its visible rays and only the long 
infra-red rays pass through. If these be brought to a focus by 
means of a lens upon a piece of platinum, the platinum will become 
luminous, and the light emitted therefrom when examined by 
means of a prism shows a continuous spectrum. This elevation in 
the rank of wave-length is called calorescence. 

Section IX. 


349. Color hy absoriotion. — Color is a sensation ; it has no 
material existence. The term ''yellow light" means, pri- 
marily, a particular sensation; secondarily, it means the 
physical cause of this sensation, i.e. a train of ether-waves 
of a particular frequency. '' All objects are black in the 
dark " ; this is equivalent to saying that without light there 
is no color. 

Experiment 1. — By means of a porte-lumiere introduce a beam of 
sunlight into a dark room. With the slit and prism form a solar spec- 
trum. Between the slit and prism introduce a deep red glass; all the 
colors of the spectrum except the red are much reduced in intensity. 

It thus appears that the color of a colored transparent 
object, as seen by transmitted light, arises from the unequal 
absorption of the different colors of white light incident upon 



it. 'A red glass absorbs less red light than light of other 
colors. The color produced by absorption is rarely very 
pure, the particular hue of the transmitted light being due 
merely to a predominance of certain colors, and not to the 
absence of all others. As the absorbing layer is thicker, the 
resulting color is purer but less intense. 

Experiment 2. — We have found that common salt introduced into a 
Bunsen flame renders it himinous, and that the light when analyzed with 
a prism is found to contain only yellow. Expose papers or fabrics of 
various colors to this light in a darkened room. No one of them except 
yellow exhibits its natural color. 

Experiment 3. — Hold a narrow strip of red paper or ribbon ^ in the 
red portion of the solar spectrum ; it appears red. Slowly move it 
toward the other end of the spectrum ; on leaving the red it becomes 
darker, and when it reaches the green it is quite black or colorless, and 
remains so as it passes the other colors of the spectrum. Repeat the 
experiment, using other colors, and notice that only in light of its own 
color does each strip of paper appear of its natural color, while in all 
other colors it is dark. 

These experiments show that the color of a body seen by 
light reflected from it depends both upon the color of the 
light incident upon it and upon the nature of the body. 

If a piece of colored glass, AB (Fig. 289), be held near a 
window so as to receive, obliquely, rays of sunlight, a portion 
of the light will be reflected by the anterior surface of the 
glass, and, falling ^^ ^^ 

upon the white ceil- 
ing, will illuminate 
it with white light. 
Another portion of 
the light will enter 
the glass and be re- 
flected from the pos- 
terior surface ; this light, having entered the glass and 

1 Care must be exercised to select only pure colors. 


traveled in it a distance a little greater than twice its thick- 
ness, will suffer an unequal absorption of its rays, and after 
emerging from the glass will, if the glass be blue, illuminate 
a neighboring portion of the ceiling with blue light. This 
illustrates the method by which pigments afford color. Thus, 
the first surface of a water-color drawing reflects the white 
daylight. Most of the light reflected to the eye has, how- 
ever, passed through the pigment to the white paper beneath, 
and being reflected from this, again passes through the layer 
of pigment before reaching the eye. With less transparent 
pigments the light may be reflected merely by particles of 
pigment beneath the surface. The color of paints and pig- 
ments is, therefore, due to the rays which they absorb least 
readily. When we paint our houses we do not apply color to 
them; we apply substances which have the property of ab- 
sorbing or subtracting from white light largely all the colors 
except those which we would have our houses appear. This 
is technically called selective absorption. 

The color of bodies thus depends generally upon their mo- 
lecular structure. Different bodies quench different portions 
of the complex sunlight. The unqueiiched light determines 
the color of a body. 

The molecular action in the case of absorption is this. The 
molecules of the substance receiving incident light are capable of 
vibrating in unison with certain ether waves. The energy of cer- 
tain of the ether-waves is employed in setting up molecular vibra- 
tions in the substance, thus raising its temperature, while the 
energy of waves of different period is propagated through the me- 
dium without producing this effect. 

350. Opalescence. Sky colors. 

Experiment 4. — Dissolve a little white castile soap in a tumbler of 
water ; or, better, stir into the water a few drops of an alcoholic solution 
of mastic, enough to render the water slightly turbid. Place a black 
screen behind the tumbler, and examine the liquid by reflected sunlight, 


— the liquid appears to be blue ; examine the liquid by transmitted sun- 
shine, — it now appears yellowish red. 

Experiment 5. — Pour some of the turbid liquid into a small test-tube, 
and examine it and the tumbler of liquid by transmitted light ; the former 
appears almost colorless, while the latter is deeply colored. 

When a medium holds in suspension fine particles of mat- 
ter, the shorter light-waves are most abundantly reflected, 
giving a blue color. The blue is purer as the particles are 
smaller. Objects seen through such mediums appear of the 
complementary hue (see § 354). This phenomenon is called 
opalescence. It accounts for the blue of watery milk, opa- 
lescent glass, smoke, and the sky. 

Skylight is reflected light. The minute particles (of water, 
probably) that pervade the atmosphere, like the fine particles 
of mastic suspended in the water, reflect blue light; while 
beyond the atmosphere is a black background of darkness. 
But we must not, from this, conclude that the atmosphere is 
blue ; for, unlike blue glass, but like the turbid liquid, it 
transmits yellow and red rays freely, so that seen by re- 
flected light it is blue, but seen by transmitted light it is 
yellowish red. 

The remarkable "yellow days" of the summer of 1883 are 
explained in this way. The atmosphere on this continent 
was very turbid during those days. 

When the sun is near the horizon, its rays travel a greater 
distance in the air to reach the earth than when it is in the 
zenith (see Fig. 263) ; consequently, there is a greater loss by 
absorption and reflection in the former case than in the latter. 
But the yellow and red rays suffer less destruction, propor- 
tionally, than the other colors ; consequently, these colors 
predominate in the morning and evening. 

351. Mixing colors. — A mixture of all the prismatic colors 
in the proportion found in sunlight produces white. Can 
white be produced in any other way ? 



Experiment 6. — On a black surface, A (Fig. 290), lay two small rec- 
tangular pieces of paper, one yellow and the other blue, about two inches 
apart. In a vertical position between these papers, 
and from 3 inches to 6 inches above them, hold a slip 
of plate glass, C. Looking obliquely down through 
the glass j^ou may see the blue paper by transmitted 
light-waves and the yellow paper by reflection. That 
is, you see the object itself in the former case, and 
the image of the object in the latter case. By a little 
manipulation the image and the object may be made 
to overlap each other, when both colors will ap- 
parently disappear, and in their place the color which 
is the result of the mixture will appear. In this case 
it will be white, or rather, gray^ which is white of a 
If the color be yellowish, lower the glass ; if 

Fig. 290. 

low degree of luminosity. 
bluish, raise it. 

Experiment 7. — With the rotating apparatus, rotate the disk (Fig. 291) 
which contains only yellow and blue. The colors {i.e. the sensations) so 
blend in the eye as to produce the sensation of gray. 

Fig. 292 represents "Newton's disk," which contains the 
seven prismatic colors arranged in a jjroper proportion to 
prodnce gray when rotated. 

Fig. 291. 

Fig. 292. 

Fig. 293. 

In like manner, you may produce white 15y mixing purple 
and green ; or, if any color on the circumference of the circle 
(see Complementary Colors, Plate I) be mixed with the color 
exactly opposite, the resulting color will be white. Again, 
the three colors, red, green, and violet, arranged as in Fig. 
293, with rather less surface of the green exposed than of 


the other colors, will give gray. Green mixed Avith red, 
in varying proportions, will produce any of the colors 
in a straight line between these two colors in the diagram 
(Plate I); green mixed with violet will produce any of the 
colors between them ; and violet mixed with red gives 

All colors are represented in the spectrum, except the 
purple hues. The latter form the connecting link between 
the two ends of the spectrum. Our color chart (Plate I) is 
intended to represent the sum total of all the sensations of 
color. By means of this chart we may determine the result 
of the (optical) mixture of any two colors, as follows : Find 
the places occupied upon the chart by the two colors Avhich 
are to be mixed, and unite the two points by a straight line. 
The color produced by the mixture will invariably be found 
at the center of this line. 

352. Mixing pigments. 

Experiment 8. — Mix a little of the two pigments, chrome yellow and 
ultramarine blue, and you obtain a green pigment. 

The last three experiments show that mixing certain colors, 
and mixing pigments of the same name, may produce very 
different results. In the first experiments you mixed colors ; 
in the last experiment you did not mix colors, and we must 
seek an explanation of the result obtained. If a glass vessel 
with parallel sides containing a blue solution of sulphate of 
copper be interposed in the path of the light-waves which 
form a solar spectrum, it will be found that the red, orange, 
and yellow waves are cut out of the spectrum, i.e. the liquid 
absorbs these waves. And if a yellow solution of bichromate 
of potash or picric acid be interposed, the blue and violet 
waves will be absorbed. It is evident that, if both solutions 
be interposed, all the colors will be destroyed except the 
green, which alone will be transmitted ; thus : — - 



Cancelled by the blue solution, ^ ^ G B V. 

Cancelled by the yellow solution, K O Y G |^ y. 

Cancelled by both solutions, ]^ ^ G ^ /. 

In a similar manner, when white light strikes a mixture of 
yellow and blue pigments on the palette, it penetrates to some 
depth into the mixture ; and, during its passage in and out, 
all the colors except the green are destroyed ; so the mixed 
pigments necessarily appear green. But when a mixture of 
yellow and blue waves enters the eye, we get, as the result 
of the combined sensations produced by the two colors, the 
sensation of white ; hence a mixture of yellow and blue gives 

The color square 3 (Plate I) represents the result of the 
mixture of pigments 1 and 2 ; while 4 represents the result 
of the optical mixture of the same colors. 

353. Theories of color vision. 

Brewster (1831) presented the hypothesis that all colors are formed 
by the union of three primaries, — red, yellow, and blue, — which 
together compose white light, and give, by combinations in twos, 
the hues, orange, green, purple, etc. This hypothesis was based on 
the mixing of pigments. But the actual addition of colors does not 
give the same result as the mixing of pigments, as has been shown. 

The generally accepted theory of color-vision is that of Dr. Young 
(1801-2), verified by Maxwell and Helmholtz. It supposes the ex- 
istence of three color sensations, red, green, and violet. These ex- 
cited simultaneously, 
and with proper in- 
tensities, produce the 
sensation of white 
light. Combined in 
twos, they produce 
the remaining color 
sensations. Thus red 
and green sensations 
combined give yellow 
or orange ; green and violet give blue, etc. The longer light-waves 
excite the sensation of red ; together with those somewhat shorter, 


they excite both red and green, thus giving yellow, and so on. 
Strictly speaking, light-waves of any length excite all three sensa- 
tions ; but usually either one or two of them greatly predominate. 

The relative intensities of the various color-sensations throughout 
the spectrum as obtained from actual measurement by Maxwell are 
shown in Fig. 294. 

354. ComjjleTnentary colors. 

Experiment 9. — On a piece of gray paper lay a circular piece of blue 
paper 15 mm in diameter. Attach one end of a piece of thread to the 
colored paper, and hold the other end in the hand. Place the eyes with- 
in about 15 cm of the colored paper, and look steadily at the center 
of the paper for about fifteen seconds ; then, without moving the eyes, 
suddenly pull the colored paper away, and instantly there will appear on 
the gray paper an image of the colored paper, but the image will appear 
to be yellow. This is usually called an after-image. If yellow paper be 
used, the color of the after-image will be blue ; and if any other color 
given in the diagram (Plate I), the color of its after-image will be the 
color that stands opposite to it. 

This plienomenon is explained as follows : When we look 
steadily at blue for a time, the eyes become fatigued by this 
color, and less susceptible to its influence, while they are fully 
susceptible to the influence of other colors ; so that when they 
are suddenly brought to look at white, which may be regarded 
as a compound of yellow and blue, they receive a vivid im- 
pression from the former, and a feeble impression from the 
latter ; hence the predominant sensation is yellow. Any two 
colors which together produce white are said to be comple- 
mentary to each other. The opposite colors in the diagram 
(Plate I) are complementary to one another. 

The complement of green is purple, — a compound color 
not existing in the spectrum. 

The eye gives no direct knowledge that the composition of 
light produces the sensation of white. Whether this sensa- 
tion is produced by the coexistence of all the rays of the 
visible spectrum, by a combination of light of two comple- 


mentary colors, or by a mixture of rays of tlie three primary 
colors, can be determined only by some process of physical 

355. Effect of contrast. — When different colors are seen at 
the same time, their appearance differs more or less from that 
observed when they are seen separately. Thus a red object 
(e.g. a red rose) appears more brilliant if a green object be 
seen in juxtaposition with it. Such effects are said to be due 
to contrast. 

When any two colors given in the circle (Plate I) are 
brought in contrast, as when they are placed next each 
other, the effect is to move them farther apart in the color 
scale. For example, if red and orange be brought in con- 
trast, the orange assumes more of a yellowish hue, and the 
red more of a purplish hue. Colors that are already as far 
apart as possible, e.g. yellow and blue, do not change their 
hue, l^ut merely cause each other to appear more brilliant. 

356. Color-blindness. — In this defect in vision, one of the 
three color sensations is either wanting or deficient, usually 
that of red ; so that the colors perceived are reduced to those 
furnished by the remaining two sensations, viz., green and 
violet. This causes the red-blind person to confound reds, 
greens, and grays. In some rare cases the sensation of green 
or violet is the one deficient. 

Section X. 


We have already studied wave interference in the case of 
sound (see § 178), and must recall that two sets of sound- 
waves may neutralize each other and produce silence. As 
an example, we cite the phenomenon of beats, in which the 
alternate increase and diminution of intensity is due to the 

young's theory. 395 

interference of two sets of sound-waves in the same and oppo- 
site phases respectively. If radiation be a wave motion, 
similar phenomena ought to occur under proper conditions. 

357. Yoiong^s theory. — The earliest authentic experiments, 
on the interference of light-waves were made by Dr. Young 
in 1801. He admitted a beam of sunlight into a dark 
chamber through a very narrow aperture, and in its path 
placed a screen having two very small openings quite near 
together. When the two overlapping pencils from these 
openings were made to fall on a white screen there appeared 
a series of bands alternately bright and dark, which disap- 
peared when one of the holes was covered. 

Although Young ascribed this phenomenon to interference, 
and explained it very satisfactorily, with the wave theory as 
a basis, his views were by no means universally accepted. 

Grimaldi, nearly a century and a half before, had noticed light 
and dark fringes at the edges of shadows of small opaque bodies 
placed in the path of sunlight admitted through a small hole. This 
action was called diffraction (see § 360), and was afterwards studied 
by Newton. In view of the emission theory, the phenomenon was 
explained by assuming that the light particles experience an at- 
tractive or repellent force as they come near the edges of bodies. 
Believers in the latter theory then contended that Young's experi- 
ment was simply one of diffraction. 

To remove these objections it was necessary to devise means of 
producing the same result, but entirely independently of apertures 
and opaque bodies. This was first done by Fresnel,! who contrived 
two most ingenious experiments for producing the light and dark 
bands, m which the results could be accounted for only by assuming 
Young's theory of interference to be correct. 

Let us now consider what was observed by Young. A and B 
(Fig. 295) are two illuminated apertures very near together, and 
emitting waves in all directions. When these two sets of waves 
arrive at any point P on the screen, they will be in the same phase 
only if the distance of P from one of the sources, say B, is one or 

1 A full account of Fresnel's experiments is given in " The Theory of Light," by 
Preston, §§ 86, 87 



Fig. 295. 

more complete wave-lengths more than the distance P A. In this 
event the two waves will conspire to increase the illumination at P, 
and this point will be on a bright band. 

On the other hand, if P B - P A be equal 
to a half wave-length or any odd number of 
half wave-lengths, the waves from A and B 
will arrive at P in exactly opposite phases 
and destroy one another, and P will be a 
point on a dark band. At intermediate 
positions for P, P B - P A will not equal 
any whole number of half wave-lengths, 
and hence the waves will meet in neither 
the same nor opposite phases, and the il- 
lumination at P will be intermediate in its 
intensity between that of the brightest and 
darkest points. Thus the bands will shade off imperceptibly into 
one another. 

In the above discussion we have assumed the light to be mono- 
chromatic, i.e. all of the same wave-length. Let O M be a perpen- 
dicular to the screens half way between A and B ; then the system 

of bands evidently is 
arranged symmetrical- 
ly about M, this point 
being on a bright band. 
It is easy to see that 
the distance of any 
given band from M 
bears a simple relation 
to the wave-length, 
and hence the bands will be closer together for short waves than for 
long ones, as shown in Fig. 296. 

If composite light be used, we should expect the bands to be colored, 
which is the case for a few of them near the central band, which is 
white ; the edges nearer M are blue, while the outer edges are red. 
As each color gives rise to a separate system of bands, the red 
ones being broadest and the violet ones narrowest, it will happen 
that after a few alternations a red band and a violet one will fall at 
the same place ; or the dark spaces of one system will be filled by 
the bright parts of another. Soon, then, as we recede from M, the 
bands become less distinctly marked, and finally merge into one 
another and fade into uniform illumination. 


■awm nil 



Fig. 296. 


358. Colors of thin x^lates. — Everybody is familiar with 
the beautiful color effects produced when ordinary white light 
falls upon a thin film of a transparent substance, such as a 
soap bubble or a film of oil on water. This is a case of inter- 
ference. Some of the light comes to the eye reflected from 
the front or nearer surface of the film ; another portion has 
entered the medium and been reflected from the back surface. 
When it emerges, it has been retarded an amount depending 
on the distance it has traveled in the film. We might expect 
that if the retardation were an even number of half wave- 
lengths, the two portions of light would be in a condition to 
interfere in the same phase, and that the effect would be 
increased ; while if the retardation were an odd multiple of 
half wave-lengths, the interference would be destructive, and 
darkness would result. 

The fact is exactly the reverse of this, since by the act of 
reflection in the denser medium the phase of the wave is 
reversed and the result is as if the wave had been retarded 
another half wave-length. 

If the eye view such a film in monochromatic light, the 
portion entering the film will be retarded varying amounts 
depending on the angle of incidence and on the thickness. 
The film will, therefore, be crossed by bright and dark 

If, however, composite light be used, cooperative or de- 
structive interference evidently cannot take place for the 
different colors at the same points, and the familiar iris- 
colored bands result. 

Experiment 1. — Press firmly together two poHshed pieces of thick plate 
glass. A number of colors will be seen arranged in a certain order, and 
forming curves more or less regular around the point of pressure. Ex- 

359. ]\^ewto7i's rings. 

Newton's method of studying these colors was very simple 
and effective, and the phenomena exhibited are known as ' ' New- 


ton's rings," If a convex lens of very small curvature be placed 
firmly upon a piece of plate glass (Fig. 297) the film of air between 
the two increases in thickness from the center radially, and hence 

Fig. 297. Fig. 298. 

beautiful circular interference bands are shown, having the point 
of contact as their center (Fig. 298). 

360. Dijfraction. 

This phenomenon, first observed by Grimaldi, and already referred 
to (see §357), occurs when light passes through a very narrow aper- 
ture or close to the §dge of an opaque body. 

Newton's strong reason for rejecting the wave theory was that 
light, as he supposed, did not go round corners as sound does. 
Closer examination, however, shows that the cases are similar if 
consistent conditions be maintained. Light really does bend round 
corners, while on the other hand, well defined sound shadows may 
be cast by sounds of sufficiently short wave-length. In other words 
sound bends round corners very much more readily than light, 
merely because its wave-length is so much greater in comparison 
with the size of the obstacle. 

The true explanation of diffraction phenomena was given by 
Fresnel, who attributed them to the mutual interference of the 
secondary wavelets which diverge from the primary or main wave- 
front as it meets the obstacle or edges of the aperture ; just as the 
cases of interference previously described are due to the mutual in- 
terference of two waves. For a fuller description of the cases of 
diffraction and their treatment, the student is referred to special 
works on optics. 

361. The Diffraction Grating} 

When a distant source of light is viewed through a system of very 
narrow rectangular openings, a central image is seen, and on either 
side of it there are several highly colored spectral images, increasing 

iFor a more complete treatment of gratings, see Theory of Light, by Preston. 



in breadth but diminishing in brightness as they recede from the 
center. A device like the above is known as a diffraction grating, 
and may be produced by ruling with a diamond fine lines on a 
piece of glass. ^ Tlie lines form the opaque portions of the grating, 
while the spaces between them are the slits through which the light 
passes. The effect is most marked when the opaque parts and 
transparent parts are equal. 

A magnified section of a grating perpendicular to the Imes is 
shown in Fig. 299. Let a h represent a line and adjacent slit in 
such a position with 
reference to the point 
P, where the eye is 
placed, that P 6-P a 
equals one wave- 
length. It is clear 
that the portion of the 
incident wave (sup- 
posed to be homoge- 
neous) corresponding 
to a & may be divided 
into two nearly equal 
parts which would de- 
structively interfere at 
P if the grating were 

not present. The effect of the grating then is to intercept one part 
of these interfering portions and enable the other to become visible. 
A bright band will then be seen in the direction P h. 

The same will happen for all similar parts of the grating provided 
the distances of their extremities from P differ by a whole number 
of wave-lengths. Thus there will appear a succession of bright 
bands at increasing angular distances from P M. It should be 
noticed in studying this figure that a 6 is extremely small compared 
with P M, and therefore the figure is necessarily distorted for 

If the light incident on the grating be composite, evidently the 
angle a, indicating the direction from P M of the bright band, will 
be less for the short waves than for the long ones. Therefore in 

Fig. 299. 

iMany of the fine gratings of Eowland have 20,000 lines to the inch. These, how- 
ever, are now usually ruled on speculum metal, as glass is apt to injure the diamond 



this case the resulting band will be colored — the inner edge blue, 
the outer one red, the portion lying between, yellow and green. 

Such a phenomenon is called a diffraction spectrum. It is what 
is known as a normal spectrum^ because it exhibits the colors, or 
Fraunhofer lines which locate them, always in their true order and 
separated by spaces bearing a simple relation to their wave-lengths. 

This is not the case in spectrums produced by refraction. The 
rays at the red end are crowded together and condensed out of all 

B c 




G H 

Fig. 300. 

proportion to those at the violet end ; i.e. a given difference in wave- 
length causes a much greater separation in the more refrangible parts. 
This is called irrationality of dispersion. It is exhibited variously by 
different substances. Fig. 300 shows the positions of the principal 
Fraunhofer lines as given with prisms (1) of flint glass, and (2) of 
crown glass, having the same refracting angle. The difference 

A a 3 C 


E b F 




8|0 7|5 7|0 6|5 6|0 5\o 5|o 4J5 f 
ajO 7|5 7|0 6|5 6^ ^5. sjo ^5 4|o 


Fig. 301. 

E b 

G 7i H^H 

between a normal or diffraction spectrum and a prismatic one may 
be understood by a glance at Fig. 301. The scales between the 
spectrums indicate the wave-lengths in hundred thousandths of a 

The grating furnishes a simple means of measuring wave-length. 


Referring again to Fig. 299, in which P & is the direction of the 

first bright band, we see tliat the triangles h ak and 6PM are 

very nearly similar {a k being a very short arc practically per- 

b k & M 
pendicular to P 6). Then —r = — — = sin a ; or bk = ab sin a. 
' ab bF 

But bk is one wave-length, X, and a 6 is - where n is the number 


of lines per unit of length on the grating ;" hence X = - sin a. 

Experiment 2. — Introduce a borax bead into the flame of a Bunsen 
burner, and place them near the wall of a darkened room. Prom a 
distance of six or eight meters, view the flame through a grating 
(one having from one to two thousand lines to the centimeter pre- 
ferred), holding it perpendicular to the line of sight and with the 
rulings vertical. Note the position on the wall of the first image of 
the flame, and measure in centimeters its distance from the flame. 
This divided by the distance from the point of observation ta the 
image will be sin a. (In practice it is found to be sufficiently 
accurate to measure from the point of observation directly to the 
burner. ) If we divide sin a by the number of lines per centimeter 
on the grating, the result will be the wave-length in centimeters 
for sodium light. 

362. Reflection gratings. 

Spectrums similar to those already described may be produced 
by reflection from a polished surface (usually of speculum metal) 
finely ruled with paral- 
lel grooves. 1 

The beautiful colors 
exhibited by the pol- 
ished surface of mother- 
of-pearl, by the feathers 
of certain birds, and 
other striated surfaces, /''T~N 
are due to wave inter- (^ 1 
ference. This was V.L---' 
demonstrated in a strik- (^'rating 

r. o- T^ •/. Fig. 302. 

mg way by Sir David 

Brewster, by taking an impression of the surface in wax, when the 
indented wax showed all the colors of the original surface. 

1 Fig. 302 represents a sectional view of a grating spectroscope. 



Section XL 


363. Double refraction. — W6 have hitherto assumed that a 
ray of light incident upon a transparent body is refracted 
according to the law^ of sines. This is the case when the 
transparent body is isotropic, i.e. having the same properties 
in all directions. There are numerous transparent substances 
which fulfil this condition at least approximately, such as 
fluids, well annealed glass, etc. On the other hand, there are 
numerous transparent substances for which the law of sines 
does not hold. When a ray of light enters such a body it is 
spli^i into two rays, and this law does not hold for both these 
resulting rays ; sometimes it does not hold for either ray. 
This gives rise to a group of phenomena known by the term 
double refraction, and the substances which affect light in this 
manner are called doubly refracting substances. These include 
various crystals, animal substances 
such as horn and shells, vegetable 
substances such as resins and gums, 
and certain artificial substances such 
as jellies and un annealed glass. 

Experiment 1. — Through a card make 
a pin-hole, and hold the card so that you 
can see skylight through the hole. Now 
bring a crystal of Iceland spar i (Fig. 30.3) 
between the eye and the card, and look at 
the hole through two parallel surfaces of 
the crystal. There will appear to be two 
holes, with light shining through each. Cause the crystal to rotate in a 
plane parallel with the card, and one of the holes will appear to remain 
nearly at rest, while the other revolves around the first. 

1 This crystal, sometimes called caicite or calc-spar, is found most abundantly in 
Iceland. It exhibits the property of double refraction very clearly, and by means of 
it this property was first discovered. By cleavage it can always be brought into the 
form represented by the diagram, which is called a rhomh. 

Fig. 303. 



Fig. 301. 

A ray of light, PQ, immediately on entering the crystal is divided into 
two parts, one of which, QO, obeys the regular law of refraction ; the 
other, QE, does not. The former is called the ordinary ray ; the latter, 
the extraordinary ray. In all crystals which 
produce this phenomenon there is one direction, 
and in some there are two, in which, if an object 
be looked at through the crystal, it does not ap- 
pear double. If all the edges of a crystal of 
Iceland spar (Fig. 304) be equal, and the line con- 
necting the two opposite obtuse solid angles, AB, 
be cut near each extremity by a plane perpendicular to it, objects viewed 
in this line, or in any line parallel with it, do not appear double. 

In every direction in which, one looks through, the crystal, 
except that parallel to AB, objects seen through it appear- 
double (see Fig. 305). The line AB is called the oj^tiGaxis of 
the crystal, and is a line around which the molecules of the 
crystal appear to be arranged symmetrically. A crystal is 
called uniaxial when it has only one optic axis, and biaxial 

PATH JlslanJe^ 

Fig. 305. 

when it has two such axes. A plane parallel to this axis and 
perpendicular to one of the rhombic faces of the crystal is 
called a principal section. The two rays travel with unequal 
speeds in the crystal in all directions except in the direction 
of the optic axis of the crystal. 

The property of double refraction may be imparted per- 
manently or transiently to certain substances which do not 
naturally possess it. Glass may be given this power by 
heating different parts unequally, and also by compression 
and bending. 


364. Nicol prism. 

For certain purposes, such as are indicated in § 371, it is best to 
allow only one of the two rays to leave the prism (that, namely, in 
the direction of the incident light), and to elimi- 
nate the other. The Nicol prism consists of a 
crystal of Iceland spar divided diagonally, as 
a b (Fig. 306), the two surfaces being cemented 
together with Canada balsam. All the faces 
of the prism are painted black except the two 
end faces. The extraordinary ray, falling 
upon the transparent balsam at an angle less 
than the critical angle, passes through it, but 
the more refracted (or ordinary) ray meets the 
balsam at an angle greater than the critical 
angle,! and is therefore totally reflected, 
\j thrown to one side of the prism, and absorbed 

Fig. 306. hy the black paint. 

365. Polarizatio7i of light hy double refraction. — On exami- 
nation of the two rays resulting from splitting a single ray 
by double refraction, it is found that each is unlike a ray of 
common light, that each has properties with respect to a fixed 
direction, and that these fixed directions for the two rays are 
at right angles to each oth^er. In short, a beam of light thus 
treated is not alike upon all sides, but has certain relations to 
surrounding space other than direction. This property can 
be given to light in various ways. To this phenomenon, how- 
ever produced, has been given the name polarization.'^ 

Slices of crystal of the mineral tourmaline, cut in planes 
parallel with their axes, are prepared and sold for optical 
experiments. If two of these slices similarly situated, as in 
Fig. 307, be placed between the eye and a card pierced by a 
hole, the hole will be plainly visible. But if one of the slices be 

1 The refractive index of Canada balsam is intermediate between tbe indices of the 

2 Newton came to the conclusion that each of the two rays had two sides ; and 
fiom the analogy of this two-sidedness with the two-endedness of a magnet the term 
polarization arose. 



Fig. 307. 

Fig. 308. 

slowly rotated in a plane at right angles with the beam of 
light, the hole will grow dimmer until the slice has passed 
through a quarter of a revolution (as represented in Fig. 308), 
when it disappears. 
If the rotation be 
continued, the hole 
reappears, faint at 
first, but reaching its 
maximum brightness 
at the end of another quarter-revolution. Thus, at suc- 
cessive quarter-revolutions it is alternately extinguished and 

It appears, then, that light which has passed through one 
transparent slice of tourmaline differs so much from common 
light, that a second similar slice may act like an opaque body, 

and stop it altogether. 
The action of the tour- 
maline may be com- 
^^ib^l^s-g::^! pared to that of a gratin g 
(A, Fig. 309) formed of 
parallel vertical rods, 
which will allow all vertical planes (as a «') to pass, but stops 
the planes (as c c') that are at right angles to these rods. 
Any plane that has succeeded in passing one grating will 
readily pass a second similarly placed. But if the second 
grating, B, be turned so that its rods are at right angles to 
the first, the plane that has succeeded in passing through the 
first grating will be stopped by the second. Light in this 
condition is iMlarized ; polarization is either the act of pro- 
ducing the change in the light, or the result of the change, 
and the instrument used is a polarizer. 

In order to understand this phenomenon, it is necessary 
to know more of the undulatory theory of light. This theory 
supposes that the undulations in ether which constitute light 

Fig. 309. 



are much like undulations in a cord when one end is shaken 
by a hand ^ as seen in Fig. 310. If the hand move vertically, 

all the undulations 
will lie in a vertical 

Fig. 310. 

plane ; if the move- 
ments of the hand be 
horizontal or oblique, 
the undulations lie in corresponding planes. So we can 
produce these waves on the rope in any plane passing 
through the rope, and can change rapidly from one plane to 
another. These waves appear differently when viewed from 
different sides. It is believed that if we could look endwise 
at a ray of light for an instant, we should see the ether 
vibrations, as in the figure of the rope, in one plane ; but in 
only a thousandth of a second so many million waves reach the 

Fig. 311. 

eye that there is time for the vibrating particles, which, like 
the hand, start the waves, to vibrate in many planes. In an 
ordinary beam of light, as it reaches the eye, there are there- 
fore undulations in all possible planes, as is partially repre- 
sented by the cross section A (Fig. 311). But rectilinear 
motion may be considered as the resultant of two similar 
motions at right angles to each other. So here, for many 
practical purposes, the vibrations may be regarded as taking 
place in only two sets of planes at right angles to each other, 

iThe vibratory motion wliicli constitutes light must be transverse to the direction 
of the ray ; it cannot be in the direction of the ray, for then there would be no dif- 
ference between the different sides of the ray ; and the phenomena of polarization 
would be unexplainable on this hypothesis. 


as represented by B of the same figure. Now, when a ray of 
light consisting, according to supposition, of undulations in 
planes at right angles to one another strikes a slice of tour- 
maline, its molecular structure resolves the motion into two 
motions, one parallel to, and the other perpendicular to, its 
axis. The former of these is transmitted and the other 
is absorbed. By this means the undulations are reduced to 
those in parallel planes only, as represented in C. The un- 
aided eye cannot usually detect any difference between com- 
mon and polarized light. An instrument Avhich will enable 
the eye to detect polarized light is called an analyzer ; thus 
the first slice of tourmaline serves as a polarizer, and the 
second slice as an analyzer. A complete polarizing apparatus, 
called a polarlscope^ used for observing the phenomena of 
polarized light, consists essentially of a polarizer and an 

366. Plane of 'polarization. 

There are several ways in which it is possible to restrict the vibra- 
tions of a ray of light to one plane. The ray in such a case is said 
to be '£ilane 'polarized. 

It will be shown presently that light is partially polarized by 
ordinary reflection. The plane of incidence in which this occurs is 
called the pZcme of 'polarization. It is still an open question 
whether the vibrations are parallel to or perpendicular to this 
plane, the evidence, however, being decidedly in favor of the latter 

With a uniaxial doubly-refracting -crystal {e.g. Iceland spar), the 
ordinary ray is polarized in a plane containing the incident ray and 
the optic axis. 

367. Polariscope consisting of two Nicol prisms. 

When a ray of light becomes split by double refraction, each of 
the resulting rays is found to be plane polarized, one being polarized 
in the plane of incidence, the other at right angles thereto. 

A and P (Fig. 312) represent two corks having Nicol prisms ex- 
tending through them lengthwise, P serving as a polarizer, and A 



as an analyzer. If the analyzer A be turned so that its principal 
section (see § 363) of d' E is pEirallel to the principal section c d R of 
the polarizer P, the ray R which enters P as unpolarized light be- 
comes polarized and the extraordinary ray emerging from this 
prism will pass freely through the analyzer A. The same happens 

if A be turned through an angle of 180° so as to bring the same 
planes parallel again. But if A be adjusted so that its principal 
section is at right angles to that of the polarizer (as in the lower part 
of Fig. 312), then the ray is quenched by the prism A. No light 
leaves the analyzer, and accordingly, to an eye placed at the end of 
A, the field of vision is dark. The same happens if A be turned 
through an angle of 180°. In all cases where the principal sections 
are neither parallel nor at right angles, the polarized light entering 
the analyzer is separated into ordinary and extraordinary rays, and 
the light which emerges from the analyzer varies in intensity with 
the angle at which the principal sections of the two prisms are in- 
clined to each other. 

368. Polarization by reflection. 

Malus discovered (1808), while looking through a double-image 
prism at the light reflected from a window in the Luxembourg 
palace in Paris, that light may be polarized by reflection. 

It was subsequently found that the amount of polarization de- 
pends on the incident angle. The angle at which the polarization 
is a maximum is called the angle of polarization by reflection, which 
in the case of glass is between 55° and 56°. Then, of course, if a 



Nicol prism be held in certain positions the reflected light will pass 
through it, but at a distance of 90° it refuses to pass. Again, light 
which has been polarized by reflection from one glass surface, A 
(Fig. 313), will be reflected from another glass surface, B, placed so 

Fig. 313. 

that the plane of incidence of the polarized ray is parallel to the 
plane of polarization, but utterly refuses to be reflected when the 
plane of incidence is at right angles to the plane of polarization, 
as in B'. So that in polariscopes a plate (better a bundle of plates) 
of glass may be substituted for either the polarizer or the analyzer. 
Light reflected obliquely from non-metallic smooth surfaces, such 
as water, polished furniture, oil paintings, etc., is found on examina- 
tion to be partially polarized. Sky-polarization is due to plane- 
polarization effected by reflection from very small particles of water 
in the atmosphere. 

369. Polarization hy refraction. 

Not only the reflected portion of the light is polarized, but also 
the part that enters the medium and is transmitted. For example, 
if A (Fig. 313) be a transparent glass plate, a part of the incident 
light will of course pass through, and this part will be found to be 
partially polarized, but in a plane perpendicular to the plane of in- 
cidence. Further, Arago found that the reflected beam and the 
transmitted beam contain the same amount of polarized light. If 
the transmitted beam be examined by making it incident upon a 
second transparent surface, it will be reflected only if the plane of 
" incidence is made parallel to the plane of polarization of the light. 


370. Circular and elliptical polarization. 

In all polarization thus far treated, we have assumed the vibra- 
tions to be in straight lines and all confined to one plane. It can 
be shown, 'however, that if two such waves, one a quarter of a 
period in advance of the other, be compounded, with their planes 
perpendicular to each other, the result will be a wave in which 
the ether particles move in circles or ellipses, according as the 
amplitudes of the components are equal or not. Such a wave is 
circularly or elliptically polarized, and since (as in all wave motion) 
the particles move successively, the wave has the form of a helix. 
For fuller discussion of this form of polarization and the methods of 
producing and detecting it, the student is referred to special works 
on light. 

371. Chromatic phenomena} 

Take the simplest case, in which the polarizer and analyzer are 
sections of tourmaline cut parallel to the axis. (A little considera- 
tion will show, however, that the explanation holds good for any 
other form of analyzer and iDolarizer.) P (Fig. 314) represents the 
polarizer and A the analyzer. P and A are supposed to be crossed. 
The arrow shows the direction of the ray. The symbols at L, B, 
C, D, are intended to indicate the condition of the light in those 
positions. L indicates the two plane component waves of which 
ordinary light may be considered as composed. One of these is the 
plane of the paper, the other a plane at right angles to it, and con- 
taining the ray. The action of the polarizer P is to remove one of 
these components, producing at B a plane-polarized wave whose 
plane coincides with the plane of the paper. This wave, lying in a 
plane at right angles to those waves which the analyzer, when in 
the position supposed, will allow to pass, is cut off, and a dark field 

results. Any wave not parallel 
to the plane of B would evi- 
dently pass through A, either 
== wholly or in part. Suppose 
now any doubly-refracting sub- 
S ^ stance, as a crystalline film S, 

Pj(j 314 to be placed between P and A. 

The plane-polarized ray B, ex- 
cept in certain special cases, is doubly refracted and separated into 

iln tlie explanation of these phenomena we have adopted largely the language of 
Prof. Cross of the Mass. Inst, of Technology. 



two component plane-polarized rays, indicated, by C, with their 
planes of vibration at right angles to each other, and inclined to the 
original plane of vibration of B. These rays at C, in which the 
vibrations are not parallel to B, and hence not at right angles to the 
axis of the tourmaline section A, are not wholly cut off, but, as A 
is a doubly-refracting crystal, are again each separated into two 
sets of rays ; one set of these having its plane of vibration at right 
angles to B and hence parallel to the axis of A, is allowed to pass 
as indicated by D, while the other set is cut off. This explains the 
illumination of the field. 

' ' The light transmitted under these circumstances is generally 
colored, for the following reason : In traversing the doubly-refract- 
ing crystal S, the components into which B is separated travel with 
different velocities. Hence one of these components gains on the 
other by an amount which, other things being equal, depends on 
the thickness of S. Suppose the gain to be such that on emerging 
from S one of the components of C is in advance of the other by 
one wave-length of red light, as shown at 0' (Fig. 315). Assuming 
a tourmaline section or equivalent analyzer to be used, one compo- 
nent only of each pair into which and 0' are resolved passes through 
A, and these components (e e\ Eig. 315) are in opposite phases, and 
hence interfere. The components o o' are not transmitted by the 
analyzer. The red rays are therefore struck out from the white 
light of the original beam, and the field appears of a greenish color. 
Evidently, if the thickness of S (Fig. 314) be such that one system 
gains upon the other an amount 
equal to any whole number of wave- 
lengths or even number of half 
wave-lengths of light of any parti- 
cular color, that color will disappear 
from the transmitted rays at D. If 
the gain be equal to any odd number 
of half wave-lengths, the compo- 
nents emerging from the analyzer 
will have the same phase, and that 

color will be present in the trans- ^ 

.^^ , , 1 Fig. 315. 

mitted beam.i 

"If a crystal of Iceland spar be substituted for the tourmaline 

analyzer A, both systems of rays, D (Fig. 314) and the set at right 

1 Read in connection with the last two statements Glazebrook's Physical Optics, 
p. 374 ; also Lloyd's Undulatory Theory of Light, pp. 219, 220. 




angles to D, will be transmitted. In this case the colors of the two 
sets of transmitted rays are always complementary ; that is, if one 
be red, the other is green ; if one be blue, the other is yellow, etc. 
This is evident from an examination of the component plane-waves 
issuing from the doubly-refracting crystal of Iceland spar at A 
(Pig. 315). It will be seen that when those of one set meet crest 
to trough, those of the other set will meet crest to crest. Hence 
any color struck out in one set will predominate in the other ; that 
is, the colors of the two sets must be complementary. Rotating the 
analyzer through 90° changes each color to the complementary 

Fig. 316. 

372. Description of a simple polariscope. 

T> (Fig. 316) is a plate of glass about 15 cm square, used as a 

polarizer. A is the ana- 
lyzer, — preferably a Mcol 
prism, — so placed as to 
view the center of the glass 
at the proper polarizing 
angle (about 55°). The 
prism, mounted in a cork, 
should be free to rotate in 
its support. S is a piece of 
ground glass used to cut off 
the images of outside ob- 
jects. G is a glass shelf, on which objects to be examined are 
placed. The instrument, covered with a black cloth, is placed on a 
table with S toward a window. 
Experiment 2. — Place on the support G a thin film of selenite or mica, 
and slowly rotate the analyzer. A beautiful display of colors will appear. 
At a certain point they will appear of maximum brilliancy, then they 
will gradually fade away and change into their complementaries. (See 

373. JUng-sijstem of plates perpendicular to axis. 

A different class of appearances is presented when a plate of any 
uniaxial crystal is examined in a convergent or divergent pencil of 
plane-polarized light. Since the rays are oblique, the thickness of 
the plate traversed increases with the angle of -incidence, being least 
for a ray parallel to the axis and normal to the plate. In this case 



the ray passes through the plate parallel to the optic axis, and is 
not doubly refracted. Let E be the position of the eye immediately 
behind the analyzer (not shown 
in Fig. 317), S S' the plate, and 
OE the direction of the axis. 
Now an oblique ray that reaches 
the eye from any point P has 
been doubly refracted in passing 
through the plate, Pa and P 6 
representing the component mo- 
tions. As the speeds of the 
component rays are not equal, 
a retardation which is greater 
as P is farther from will occur 
in- the case of one of these. 

In general, the analyzer again 
subdivides the rays P a and P b 
into two mutually perpendicu- 
lar sets, one component only of each set being transmitted. These 
latter interfere more or less destructively according to the retarda- 
tion that has been suffered by one of them, the interference being 
complete for any color when the difference of phase is half a wave- 
length for this color. It is clear, then, that the field will be 
occupied by a series of concentric rings spectral colored from 
the center outward, the appearance being similar to that of 
Newton's rings. 

Let us now consider two planes through E, one parallel to the 
plane of primitive polarization and the other perpendicular to it. 
Let their traces on S S' be M M' and N N'. Now all rays travers- 
ing the plate in these planes will be unaffected, since the principal 
section of the crystal in the one case coincides with the plane of 
polarization of the incident signs, and in the other case is perpen- 
dicular to it. 

* These rays, therefore, will or will not be transmitted by the 
analyzer according as its principal section is parallel to or perpen- 
dicular to the primitive plane of polarization. In the former case 
we shall have a white cross over the ring system, in the latter a 
dark one. 

A and B (Fig. 318) illustrate these phenomena only in a very 
imperfect way, since the coloring of the figures is necessarily 



374. Rotatory polarization. 

It was observed by Arago in 1811 that if plane polarized light be 
transmitted by a plate of quartz cut perpendicular to the axis, the 

plane of polarization is rotated 
through a certain angle. The 
investigations of Biot showed 
this angle to be proportional to 
the thickness of the plate and 
approximately to the inverse 
square of the wave-length. 
For example, if such a plate of 
quartz be placed between two 
crossed nicols, the dark field 
immediately becomes bright, 
and also colored, if the light 
employed be not homogeneous. 
That is, the plate has turned 
the plane of polarization of the 
light, so that when it is inci- 
dent upon the analyzer there 
is a component of the vibration 
which is parallel to the princi- 
pal section, and so can be trans- 
mitted. As the different colors 
are rotated through different 
angles, the transmitted compo- 
nents are not mixed in the 
same proportions as in the in- 
cident light, and so the field is 
colored, the particular color 
Fig. 318. depending on the position of 

the analyzer. It was found that some specimens of quartz cause 
rotation to the right {i.e. clockwise), and so are called dextrogyrate, 
while others cause rotation to the left, and are called levogymte. 

Many other substances, including some liquids and solutions, 
also possess this property, though to a much less degree. Thus, 
while a plate of quartz 1 mm thick rotates the plane of red light 
nearly 18°, the same thickness of turpentine produces a rotation of 
only a quarter of a degree. 

The method of determining this angle is extremely simple. So- 
dium light is commonly used, and the polarizer and analyzer are 



so adjusted that the field is dark. The substance to be examined 
is then inserted between them, and the analyzer is turned till the 
field is again dark ; the angle thus turned through is the rotation 
produced by the substance. Liquids and vapors are studied by 
enclosing them in a tube of known length through which the polar- 
ized light is passed. 

These phenomena are explained by supposing that when a beam 
of plane polarized light is incident upon the quartz plate it is 
doubly refracted, or divided, not into two plane-polarized beams, 
but into two circularly-polarized beams, the directions of motion 
being opposite. If now these traverse the crystal with slightly 
different speeds, it is clear that when, on emerging, they are com- 
pounded, the resultant plane of motion and hence plane of polari- 
zation will not be parallel to that of the incident light. Evidently 
it will have been turned through an angle depending on the thick- 
ness of the crystal and on the difference of speeds of the two beams 
in it. 

In Fig. 319 is shown the tourmaline tongs, a simple form of po- 
lariscope. Two plates of tourmaline, cut as described in § 365, 
are mounted so as to turn in eyes formed at the extremities of the 
looped wire. If the plates be so arranged that the light is com- 

FiG. 319. 

pletely extinguished, and a pebble (quartz) spectacle lens be placed 
between the tourmalines thus arranged, the light will again pass, 
showing the effects of rotation of the plane of polarization. This 
is accepted as a test of the genuineness of quartz lenses. 

375. The saccharimeter. 

The angle of rotation of a saccharine solution varies with the 
number of grams of sugar in a cubic centimeter of the solution. 
On this principle is constructed a polariscope, called a saccharimeter, 
which is used for the express purpose of determining the percentage 


of pure sugar in an aqueous solution, and hence the commercial 
value of a syrup. This instrument is furnished with a scale em- 
pirically graduated so that the percentage can be read directly, or 
easily calculated. 

Section XII. 


376. Heat not transmitted by radiation. — We have learned 
that heat may travel through matter (by conduction), and 
with matter (by convection), and it is sometimes stated that 
there is a third method by which it travels, viz. " radiation." 
Heat itself is not transferred by radiation at all ; heat gen- 
erates radiation (i.e. ether waves) at one place, and radiation 
produces heat at another; it is radiation which travels, not 
heat. It does not exist as heat in the intervening space, and 
therefore does not necessarily heat the substance filling that 
space. Heat can flow only one way, viz. from a given point to 
a point that is colder ; radiation travels in all directions. The 
sun sends us no heat, but it sends radiations which the earth 
transforms into heat ; but it should be borne in mind that 
while it is radiation it is not heat, and vice versa. Tempera- 
ture is a condition of bodies, not of radiations ; wave-lengths 
belong to radiations, not to heat which produces them. 

377. Diathermancy and athermancy. — What becomes of 
radiations which strike a body depends largely upon the 
character of the body. If the nature of the body be such 
that its molecules can accept the motion of the ether, the 
vibrations of ether are said to be absorbed by the body and 
the body is thereby heated, i.e. the undulations of ether are 
transformed into molecular energy or heat. Glass, for in- 
stance, allows the sun's radiations to pass very freely through 
it, and very little is transformed into heat. But if the glass 
be covered with the soot of a candle flame, the soot will 


absorb the radiations and the glass become heated. Observe 
how Gohl window-glass may remain, while radiations pour 
through it and heat objects in the room. Only those radia- 
tions that a body absorbs heat it ; those that j^ccss through it do 
not affect its temperature. 

Bodies that transmit radiations freely are said to be dia- 
thermanous, while those that absorb them largely are called 
athermanous. These terms bear the same relation to the 
transmission of radiant energy of any and all wave-lengths as 
do transparency and opacity to the transmission of light or 
visible radiations. The most diathermanous substance known 
is rock salt. A solution of iodine in carbon bisulphide 
absorbs almost completely the rays of the visible spectrum, 
but transmits almost completely all of longer wave-length 
than the red end of the spectrum. A plate of alum acts in 
the reverse manner, transmitting the visible and absorbing 
the invisible. Among liquids carbon bisulphide is exception- 
ally transparent to all forms of radiation ; while water, trans- 
parent to short waves, absorbs the longer waves, and is thus 
quite athermanous. 

Experiment 1. — Bring the bulb of an air thermometer into the focus 
of a burning-glass exposed to the sun's rays. The radiation concentrated 
on the enclosed air scarcely affects this delicate instrument. 

Experiment 2. — Cover the outside of the bulb of the air thermometer 
with lamp-black and repeat the last experiment. The lamp-black 
absorbs the radiant energy, and the heat conducted through the glass to 
the enclosed air raises its temperature and causes it to expand and 
rapidly push the liquid out of the stem. 

Dry air is almost perfectly diathermanous. All of the 
sun's radiations that reach the earth pass through the atmos- 
phere, which contains a vast amount of aqueous vapor. This 
vapor, like water, is comparatively opaque to long waves : 
hence it modifies very much the character of the radiations 
which reach the earth. This fact, together with what we have 



learned from Exp. 1, enables ns to understand the method by 
which our atmosphere becomes heated. Eirst, that portion 
of the radiant energy which comes to us from the sun in the 
form of relatively long waves is stopped by the watery vapor 
in the air^ which is thereby heated. The portion that comes to 
us in short waves escaping this absorption heats the earth by 
falling upon it. The warmed earth loses its heat, — partly by 
conduction to the air, still more largely by radiation out- 
ward. The form of radiation, however, has been greatly 
changed; for now, coming from a body at a low temperature, 
it is chiefly in long waves that the energy is transmitted ; 
while, as we have seen, it was largely in the form of short 
waves that the earth received its heat. But it is exactly 
these long waves which are most readily stopped by the atmos- 
phere ; hence, the atmosphere, or rather the aqueous vapor 
of the atmosphere, acts as a sort of trap for the energy which 
comes to us from the sun. 

Kemove the watery vapor (which serves as a " blanket '^ to 
the earth) from our atmosphere, and the chill 
resulting from the rapid escape of heat by 
radiation would probably put an end to all 
animal and vegetable life. Glass does not 
screen us from the sun's heat, but it can very 
effectually screen us from the heat radiated 
from a stove or any other terrestrial object. 
Glass is diathermanous to the sun's radiations 
(simply because they have already lost most 
of the very long waves by atmospheric ab- 
sorption), but quite athermanous to other 
radiations. This is well illustrated in the 
case of hot-beds and green-houses. The 
sun's rays pass through the glass of these 
enclosures almost unobstructed, and heat the 
Fig. 320. earth ; but the radiations given out in turn 


by the earth are such as cannot pass out through the glass, 
and hence the heat is retained within the enclosures. 

378. The ixuliometer. 

Fig. 320 represents an instrument called a radiometer. The 
moving part is a small vane resting on the point of a needle. It is 
so nicely poised on this pivot that it rotates with the greatest free- 
dom. To the extremities of each of the four arms of the vane are 
attached disks of aluminum or mica which are light on one side 
and black on the other. The whole is enclosed in a glass bulb from 
which the air is nearly exhausted, i 

Exposed to the radiations of the sun, a candle flame, or even the 
radiations from the human body, the vane will rotate with the un- 
blackened faces in advance. The blackened faces absorb the 
radiant energy and become heated, the air particles remaining in the 
bulb by striking against them have their speed increased, and thus 
results an increased pressure upon the blackened surfaces, causing 
a more or less rapid revolution of the arms. 

It can be shown that the glass bulb tends to rotate in the oppo- 
site direction to that of the vane. This is a proof that the rotation 
is due to action and reaction between the vane and the glass, and 
not, as it might appear, between the vane and the source of radia- 
tion. The radiometer thus serves indirectly to transform radiant 
energy into mechanical work, and may be used to measure the 
mechanical effects of radiant energy. For example, the radiations 
. from two candle flames produce twice the effect of that from one ; 
and when the distance from the source of radiations is doubled the 
effect is one-fourth as great.^ Hence the radiometer may be used 
to verify in a very direct and simple manner the law of inverse 
squares as applied to radiant energy. It is also used in a variety 
of experiments to illustrate the mechanical effects of the rapidly 
moving molecules of gases. 

If the opening at the top be kept open and connected with a 
pump, so that the exhaustion can be regulated at will, then after 
a certain degree of exhaustion has been attained the black disks 
exposed to radiation are repelled and rotation ensues. If the 

1 The bulb must be exhausted of air till the mean free path of the air particles is 
greater than the distance of the glass from the surface of the vane, so that particles 
after impinging on the disks do not as a rule collide with other particles before 
reaching the glass. 



exhaustion be gradually increased, a maximum speed is reached ; 
further exhaustion diminishes the speed, and ultimately rotation 

379. Solar radiation. Pyrheliometer. 

Fig, 321 represents an instrument, called a pyrheliometer^ used 
to determine solar radiation. It consists of a shallow cylindrical 
vessel, A, of thin metal. The upper surface is covered with lamp 
black. The bulb of a thermometer is 
enclosed in this cylinder and its stem in 
a tube, B ; the remaining space in cylin- 
der and tube is filled with water. The 
blackened surface is turned toward the 
sun, and to ensure that the rays are 
normal to this surface the shadow of the 
cylinder is made to cover exactly and 
coincide with the disk C. 

First the instrument, sheltered from 
the sun, is permitted to radiate its heat 
into the clear sky for (say) five minutes. 
Let the fall in temperature be r°. 

Next it is turned to the sun for five 
minutes. Let the rise in temperature be 

Finally it is allowed {at its increased 
temperature) to radiate into the clear sky 
as before for five minutes. Let the fall of temperature be r'°. 

Now since r denotes the change of temperature during radiation 
into clear sky before heating, ^d r' tho same after heating, the 
change of temperature during the heating, due to radiation, con- 

r -\- r' 
duction, etc., will be very nearly a mean between the two, or — - — ; 

but it is evident that this cooling effect takes place even when the 
instrument is receiving the sun's radiations, and tends to diminish 
the heating effect produced by these radiations. Hence, therefore, 
the whole heating effect will be 

r -\- r' 

Fig. 321. 

This is the number of degrees which the sun's radiant energy is 
able to raise the temperature of n kilograms of water (suitable al- 


lowance being made for the water equivalent of the vessel and 
thermometer) when it falls upon a units of area of lamp black for 
m units of time. Then the radiant energy received per unit area 
during one unit of time is equivalent to 


7^ -I- y' ~ 

H -— ) ^ (a X m) 

calories of heat. By Joule's equivalent we are able to calculate the 
energy in mechanical units. 

It is estimated that if the whole radiation received by the earth 
from the sun were employed in melting ice, it would in a year melt 
a layer of ice all round the earth 137 feet in thickness ( Young). i 
Only a very small fraction of the sun's radiations are intercepted 
by the earth. Lord Kelvin estimates that the total energy emitted 
by the sun is at the rate of 7000 horse-power per square foot of 
radiating surface. 

380. All bodies radiate heat. — Hot bodies usually part with 
their heat much more rapidly by radiation than by all other 
processes combined. But cold bodies, like ice, emit radia- 
tions even when surrounded by warm bodies. This must be 
so from the nature of the case, for the molecules of the coldest 
bodies possess some motion, and being surrounded by ether 
they cannot move without imjjarting some of their motion to 
the ether, and to that extent becoming themselves colder. 

381. Prevosfs theory of exchanges. — Let us suppose that 
we have two bodies, A and B, at different temperatures, — A 
warmer than B. Kadiation takes place not only from A to B, 
but from B to A ; but, in consequence of A's excess of tem- 
perature, more radiation passes from A to B than from B to 
A, and this continues until both bodies acquire the same 
temperature. At this point radiation by no means ceases, 
but each now gives as much as it receives, and thus equilib- 
rium is kept up. This is known as "Prevost's Theory of 

1 For further information concerning solar radiation, see Young's Elements of 
Astronomy, pp. 147-153. 


382. Good absorbers, good radiators. 

Experiment 3. — Select two small tin boxes of equal capacity, — one 
should be bright outside, while the other should be covered thinly with 
soot from a candle flame. Cut a hole in the cover of each box large 
enough to admit the bulb of a thermometer. Fill both boxes with hot 
water, and introduce into each a thermometer. They will register the 
same temperature at first. Set both in a cool place, and in half an hour 
you will find that the thermometer in the blackened box registers several 
degrees lower than the other. Then fill both with cold water, and set 
them in front of a fire or in the sunshine, and it will be found that the 
temperature in the blackened box rises more rapidly. 

As bodies differ widely in their absorbing power, so they 
do in their radiating power, and it is found to be universally 
true that good absorbers are good radiators, and bad absorbers 
are bad radiatoi^s. Much, in both cases, depends upon the 
character of the surface as well as of the substance. Bright, 
polished surfaces are poor absorbers and poor radiators ; 
while tarnished, dark, and roughened surfaces absorb and 
radiate rapidly. Dark clothing absorbs and radiates more 
rapidly than light clothing. 

383. Deta. ■ — It requires no elaborate experiments to show 
that some bodies radiate more rapidly than others. All 
nature testifies to this, every still, cloudless summer night. 
During the day objects on the earth's surface gain more heat 
by radiation than they lose, but as soon as the sun has set 
this is reversed. Then everything begins to cool by radia- 
tion into space. Objects becoming cool, the air in contact 
with them becomes chilled ; its watery vapor condenses, and 
collects in tiny liquid drops on their surfaces. But these 
dew-drops collect much more abundantly on certain things, 
such as grasses and leaves, than on others, such as stones and 
earth. The reason that dew does not collect on the latter so 
freely, is because of their poor radiating power ; they do not 
get cool as rapidly. 



1. What objections can you raise to the term " radiant heat " ? 

2. Explain why the temperature of a hotbed is above that of the sur- 
rounding air, 

3. How could you separate the dark radiation of an electric arc lamp 
from tlie luminous radiation ? 

4. How can you demonstrate the existence of ether waves of greater 
length than the light-giving waves ? 

5. Ice appears to radiate cold. Explain the phenomenon by Provost's 

6. A radiometer placed near ice rotates with the blackened sides of its 
disks in advance. Explain. 

7. What parts of the spectrum are invisible to the eye ? 

8. How can you prove tlie existence of invisible solar rays ? 

9. On what does the color of bodies primarily depend ? 

10. What agency does a body perform in determining its own color 
when illuminated with white light ? 

11. a. Why is grass green ? b. Snow, white? c. Soot, black? 

12. How does a spectrum produced by a crown glass prism differ from 
a spectrum produced by a flint glass prism ? 

13. State some phenomenon which the undulatory theory alone is 
competent to explain. 

14. Describe the appearance which an iridescent soap bubble would 
present in a monochromatic light. 

15. Objects seen across the top of a hot stove appear unsteady and 
indistinct. Explain. 

16. State how with a pair of tourmaline tongs you may distinguish a 
glass spectacle lens from a quartz lens. 

17. State how the sensation of purple is produced. 

18. What is meant by the artist's three primary pigments ? 

19. Describe the surface which a hot-water vessel should have in order 
to retain its heat well. 

20. Suppose beams of sunlight enter a dark room through two aper- 
tures in a shutter, and a blue glass be placed in the path of one beam 
and a yellow glass in the path of the other, a. What color will that 
portion of a white wall appear where the two images of the sun over- 
lap ? b. State the result if one of the apertures be closed and the beam 
from the other aperture be made to pass through both colored glasses. 

21. When red and green sensations coexist what is the resulting sen- 
sation ? 



22. What phenomenon shows that ether- waves do not traverse all 
substances with equal speed ? 

23. State how light may "turn a corner." 

24. What utility is there in keeping certain parts of a steam engine 
very bright ? 

Section XIII. 


384. Coiwpoiind microscope. — When it is desired to magnify 
an object more than can be done conveniently and v^ith 
distinctness by a single lens, two convex lenses are used, — 
one, (Fig. 322), called the objective, to form a magnified 
real image a' V of the object a h ; and the other, E, called the 
eye-jjiece, to magnify this image so that the image a' b' appears 
of the size a" b" . Instead of looking at the object as when 
we use a simple lens, we look at the real inverted image, a^b', 
of the object. 

This represents the simplest 
b" possible form of the compound micro- 

,/| scope. In practice, however, the 

^,;;''' i construction is more complicated. 

Fig. 323 represents a perspective 
and a sectional view of a simple form 
of a modern compound microscope. 
The body of the instrument consists 
of a series of brass tubes movable one 
within another. In the upper end H 
is the ocular or eye piece. It consists 
of two plano-convex lenses o and n, 
the former called the eye-lens, the 
latter called the field lens. The ad- 
use of two lenses in the eye-piece 

Fig. 322. 

vantages derived from the 
are as follows: 

1. The combination diminishes spherical aberration and thereby 
increases the flatness of the field. The images a' b' and a'' b" (Fig. 323) 
are in reality curved in consequence of the spherical aberration 



caused by the objective. The effect of the field lens is to correct 
this curvature in a measure. 

2. The combination increases the field of view, so that a larger 
area of the object is made visible at the same view. 

3. The combination diminishes chromatic aberration. 

All microscopes, however, should be furnished with an achro- 

FlG. 323. 

matic objective. This consists of two to four achromatic lenses, 
(the achromatic triplet, the most common form, is represented on 
an enlarged scale at L in Fig. 323), combined so as to act as a 
single lens of short focus. By the use of several lenses, the aber- 
rations can be better corrected than with a sinsrle lens. 



The object to be examined is placed on a stage, S, and, if 
the object be transparent, it is strongly illnminated by focus- 
ing light upon it by means of a concave mirror, M. If the 
object be opaque, it is illuminated by light directed upon it 
obliquely from above by the converging lens N. 

385. Oculars. 

The negative (or Huyghenian) ocular consists of two convex 
lenses of crown glass, F and E (Fig. 324), the convex surfaces 
being turned toward the object glass. A pencil of rays from 

the object-glass converg- 
ing towards a focus, a, is 
brought to a focus, a\ 
half way between the two 

This ocular is called 
negative because it is adapted to rays already converging. The 
focal length of F is three times that of E, and the distance between 
the lenses is one-half the sum of the focal lengths. 

The positive (or Ramsden) ocular consists of two plano-convex 
lenses, E and E' (Fig. 325), with the convex surfaces turned 
towards each other. These lenses are of equal focal length, and 
the distance between 
them is two-thirds the 
focal length of one of 
them. This combi- 
nation is not achro- 
matic. It is used 

Fig. 324. 

Fig. 325. 

when spider lines are placed in the focus of the field lens for pur- 
poses of exact measurement. 

In obtaining high magnifying power, it is generally best to use 
objectives of short focal length rather than oculars of high power, 
as the latter magnify the imperfections of the former. 

386. Magnifying power.- — The magnifying power of a com- 
pound microscope is the product of the respective magnifying 
powers of the object-glass and the eye-piece; that is, if the 
first magnify 20 times and the other ten times, the total 
magnifying power is 200. The magnifying power is deter- 


mined experimentally by means of a micrometer scale, for a 
description of which the student is referred to technical works 
on microscopy. 

387. Telescopes. — Telescopes are used to view (scope) ob- 
jects afar off (tele). They are classified as astronomical or 
terrestrial, according as they are designed to be used in view- 
ing heavenly bodies or terrestrial objects ; reflecting or re- 
fracting, according as the objective is a concave mirror or a 
converging lens. The terrestrial telescope differs from the 
astronomical in producing images in their true position with- 
out inversion. This is effected by means of an extra object 
lens, which corrects the inversion of the main object lens. 
The matter of inversion is of little or no consequence in 
viewing heavenly bodies. 

The refracting astronomical telescope consists essentially, 

Fig. 326. 

like the compound microscope, of two lenses. The object- 
glass (0, Fig. 326) forms a real diminished image (a h) of the 
object A B ; this image, seen through the eye-glass E, ax3pears 
magnified and of the size c d. The object-glass is of large 
diameter, in order to collect as much light as possible from a 
distant object for a better illumination of the image. 

This telescope is analogous to the microscope, but the two in- 
struments differ in this respect : in the microscope, the object being 
very near the object-glass, the image is formed much beyond the 
principal focus, and is greatly magnified, so that both the object- 
glass and the eye-piece magnify ; while in the telescope, the heavenly 
body being at a great distance, the incident rays are practically 



parallel, and the image formed by the object-glass is much smaller 
than the object. The only magnification which can occur is pro- 
duced by the eye-piece, which ought therefore to be of high power. 
The magnifying power of this instrument i equals approximately 
the focal length of the object-glass divided by the focal length of the 

388. The Newtonian reflecting telescope. 

For an instrument of moderate cost, specially adapted to school 
and college use owing to the ease of manipulation and the comfort 
with which the observer may view any part of the sky, the New- 
tonian reflecting telescope meets with much favor. It also possesses 
the great advantage of giving a colorless image of bright objects, 

Fig. 327. 

which cannot be obtained in a refractor. 2 Fig. 327 represents a 
horizontal sectional view of this instrument. Incident rays are 
reflected from the parabolic mirror, M ; striking the rectangular 
prism, m n, they undergo total reflection, and form at a 6 a small 
image of the heavenly body. The image is viewed through an eye- 
piece inserted in the side of the telescope. The reflector serves as 
an object-glass, and is of course free from chromatic aberration, 
while spherical aberration is corrected by the shape given the 

1 The student may ascertain the magnifying power of a terrestrial telescope by 
viewing a scale directly with one eye, and its magnified image as seen through the 
telescope with the other eye. (See the author's Laboratory Manual and Note Book.) 

2 On the whole, however, if the matter of expense be disregarded, the balance of 
advantage is generally considered to lie with the refracting telescope. Briefly, its 
chief advantages are : (1) it gives a brighter image than a reflector of the same size ; 
(2) it gives a better definition ; (3) the lens does not deteriorate with age as does the 
speculum in a reflecting telescope. 

The largest refracting telescope that has thus far been made (except the Yerkes 
telescope at the University of Chicago) is that of the Lick Observatory in California 
(see Plate in). 


Plate HL 



389. Photograypher^ s camera. — The jjJiotographer^s camera 
or camera ohscura, of which A B (Fig. 328) represents a ver- 
tical section, consists of a dark box painted black on the 

Fig. 328. 

interior. A screen of ground glass, S, forms a partition in 
the box. A sliding tube, T, contains a convex lens, L. If 
an object, D, be placed some distance in front, and the dis- 
tance of the lens from the screen be suitably adjusted, a 
distinct, real, and inverted image can be seen upon the screen 
by looking through the aperture C. When the image is prop- 
erly focused, the photographer replaces the ground-glass plate 
by a sensitized plate, and by their chemical power the sun's 
rays imprint a true picture of the object on this plate. 

390. The human eye. — Fig. 329 represents a horizontal 
section of this wonderful organ, 
eye, like a watch-crystal, is a 
transparent coat 1, called the 
cornea. A tough membrane 2, 
of which the cornea is a con- 
tinuation, forms the outer 
wall of the eye, and is called 
the sclerotic coat, or "white 
of the eye." This coat is 
lined on the interior with 
a delicate membrane 3, called 
the choroid coat ; the latter con- 
sists of a black pigment, which fig. 329. 

Covering the front of the 


prevents internal reflection. The inmost coat 4, called the 
retina, is formed by expansion of the optic nerve 0. The 
muscular tissue ii is called the iris; its color determines 
the so-called " color of the eye." In the center of the iris is 
a circular opening 5, called the pu23il, whose function is to 
regulate, by involuntary enlargement and contraction, the 
quantity of light-waves admitted to the posterior chamber of 
the eye. Just back of the iris is a tough, elastic, and trans- 
parent body 6, called the crystalline lens. This lens divides 
the eye into two chambers ; -the anterior chamber 7 is filled 
with a limpid liquid, called the aqueous humor ; the posterior 
chamber 8 is filled with a jelly-like substance, called the 
vitreous humor. The lens and the two humors constitute the 
refracting apparatus. 

Experiment 1. — a. Make a model of an eye. Fill an 8-ounce flask 
with clear water (eye-ball). Cover one side with black paper having a 
round hole in it (iris and pupil). Place a slightly convex lens in front of 
the hole (cornea and crystalline lens combined ; the latter outside the 
eye-ball instead of inside). Place a candle flame (object) in front of the 
hole at a distance of about 4 feet ; catch (inverted) distinct image of the 
flame on a paper screen (retina) behind the flask. Move the candle nearer 
the flask ; the image becomes indistinct. Eestore distinctness by inter- 
posing a converging lens (remedy for long sight). 

6. Place the candle very near the lens and focus its image on the 
screen (now in a new position). Move the candle away ; the image comes 
nearer the lens, and to carry the image back to the screen you must use 
a diverging lens (remedy for short sight). 

Experiment 2. — Make two dots on paper two inches apart. Close 
the left eye, and bring the right one over the left spot. At a distance of 
about six inches the right spot becomes invisible. As you bring the 
paper nearer, the eye turns to regard the left spot ; the image of the right 
spot meantime travels noseward over the retina, until it reaches a spot 
on the retina, called the blind spot, which is not sensitive to the action 
of light-waves. This spot is where the optic nerve enters the eye. 

The eye is a camera obscura, in which the retina serves as 
a screen. Images of outside objects are projected by means 


•of the crystalline lens^ assisted by the refraction of the 
humors, upon this screen, and the impressions thereby 
made on this delicate network of nerve filaments are conveyed 
by the optic nerve to the brain. If the two outer coatings be 
removed from the back part of the eye of an ox recently 
killed, so as to render it somewhat transparent, true images 
of whole landscapes may be seen formed upon the retina of 
the eye, when it is held in front of your eye. 

With the ordinary camera, the distance of the lens from 
the screen must be regulated to adapt itself to the varying 
distances of outside objects, in order that the images may be 
properly focused on the screen. In the eye this is accom- 
plished by changing the convexity of the lens. We can almost 
instantly and unconsciously change the lens of the eye, so as 
to form on the retina a distinct image of an object miles away 
or only a few inches distant. The nearest limit at which an 
object can be placed so as to form a distinct image on the 
retina is about five inches. On the other hand, the normal 
eye in a passive state is adjusted for objects at an infinite 

The retina, on careful examination, is found to be com- 
posed in part of little elements in its back portion, which 
have received, from their appearance, the names of rods and 
cones. It is thought that these rods and cones receive and 
respond to the vibrations of ether ; in other words, that they 
co-vibrate with the undulations of the ether, and thereby we 
get our sensation of light. 

The eye is not free from spherical or chromatic aberration, 
though these are very much reduced by the action of the iris, 
which acts as a diaphragm to cut off all except the central 

391. Defects of vision. — Myopia (short-sightedness) is caused 
by the excessive length of the globe from front to back, so 
that the images of all but near objects are formed in front of 



the retina. Eemedy : use diverging lenses. Hypermetropia 
(long-sightedness) occurs when the axis of the globe is so short 
that the image of an object is back of the retina unless the 
object is held at an inconvenient distance, in which case it 
tends to become indistinct. Eemedy : use converging lenses. 
Presbyopia is due to loss of accommodation power, so that while 
vision for distant objects remains clear, that for near objects 
is indistinct. This defect is incident to advancing years, and 
is due to progressive loss of elasticity of the crystalline lens. 
Eemedy : converging lenses. Astigmatism is caused by an 
inequality in the curvature of the cornea in different meridi- 
ans, so* that when, for example, a diagram like Fig. 330 is 
held at a distance, vertical lines will 
be in- focus and horizontal lines will be 
out of focus and will appear blurred 
and indistinct, or vice versa. Eemedy: 
lenses of cylindrical curvature. But, 
for this, as well as for all other de- 
fects or troubles of the eyes, consult 
a skilled oculist, and the earlier the 

Advice to all : Do not overstrain or 
overtax the eyes, or use them in in- 
sufficient or excessive light, in flickering light such as that of 
a gas-jet, or in unsteady light such as that in a moving 
vehicle ; and avoid so far as practicable sudden changes of 
light, such as lightning flashes, etc. 

392. Stereo2oticon. — This instrument is extensively em- 
ployed in the lecture-room for producing on a screen magnified 
images of small, transparent pictures on glass, called slides; 
also for rendering a certain class of experiments visible to . a 
large audience by projecting them on a screen.^ The lime 

iFor useful information relating to the operation of projection, especially for 
scientific illustrations, see Wright's Light, and Dolhear's Art of Projecting. 

Fig. 330. 




light is most commonly used, though the electric light is pre- 
ferred for a certain class of projections. The flame of an 
oxyhydrogen blow-pipe A (Fig. 331) is directed against a 
stick of lime B, and raises it to a white heat. The radiations 

Fig. 331. 

from the lime are condensed, by means of a convex lens c, 
called the condensing lens, (usually two plano-convex lenses 
are used), so that a larger quantity of radiations will pass 
through the convex lens E, called the projecting lens. The 
latter lens produces (or projects) a real, inverted, and mag- 
nified image of the picture on the screen S. The mounted 
lens E may slide back and forth on the bar E, so as properly 
to focus the image. 



Section I. 


393. Electrification. — Certain bodies, when the conditions 
are suitable, acquire by contact and subsequent separation (or 
more readily by friction^) the property of attracting light 
bodies such as feathers, pieces of tissue paper, etc. For 
example, glass rubbed with silk, and sealing-wax or ebonite 
with woolen cloth, manifest this property by picking up 
scraps of paper, etc. Bodies in this state are said to he 
electrified or charged with electricity. 

Experiment 1. — Balance a flat wooden ruler (Fig. 332) upon the bottom 
of an inverted flask. Rub a rubber comb with a woolen cloth or draw 

it a few times through your hair (if dry) 
and place it near one end of the ruler ; 
the ruler will turn toward the comb. 

Experiment 2. — Hold the comb over a 
handful of bits of tissue paper ; the papers 
quickly jump to the comb, stick to it for 
an instant, and then leap energetically 
from the comb. The papers are first 
attracted to the comb, but in a short 
time acquire some of its electrification, 
and then are repelled. If the papers be 
pulled off from the comb, they will cling 
Fig. 332. to the hands of the operator. 

1 Possibly because in the act of rubbing more points are brought in contact. 



Fig. 333. 

394. Tiuo kinds of electrification. 

Experiment 3. — Suspend a ball of elder pith, C (Fig. 333), by a silk 
thread. Electrify a glass rod D with a silk handkerchief and present it to 
the ball ; attraction at first 
occurs, followed by repulsion 
soon after contact. Next ex- 
cite a stick of sealing-wax or 
a rubber comb with a woolen 
cloth and present it to the ball 
which is repelled by the elec- 
trified glass ; it is attracted by 
the electrified wax or rubber. 

Experiment 4. — Suspend 
in two stirrups two glass rods 
that have each been rubbed 
with silk (Fig. 334), and pre- 
sent them to each other; they 
repel each other. Suspend two sticks of sealing-wax that have been rubbed 
with flannel in the same manner; the same result follows. Now, in a 

like manner, present one of the glass rods 
and one of the sticks of sealing-wax to 
each other ; they attract each other. 

It is evident (1) that there are 
tiuo kinds or conditions of electrifi- 
cation ; or, for convenience, we 
sometimes say tiuo kiiids of elec- 
tricity ; (2) that bodies similarly 
electrified repel one another, bodies oppositely electrified attract 
one another. 

Glass rubbed with silk is said to receive a charge of vitreous 
electrification ; the wax, after being rubbed with woolen cloth, 
on the other hand, is charged with resinous electrification. 
Vitreous and resinous electrifications bear to each other 
somewhat the same relation as positive and negative quanti- 
ties in algebra ; and by arbitrary convention vitreous charges 
are said to be positive (written +E), and resinous negative 
(written — E). 

Fig. 334. 


Experiment 5. — Once more electrify a stick of sealing-wax with woolen 
cloth, and present it to the pith ball, and after the ball is repelled, bring 
the surface of the flannel which had electrified the rod near the ball ; the 
ball is attracted by it, showing that the rubber is also electrified, and 
with the opposite kind to that which the sealing-wax possesses. 

One kind of electrification is never developed alone ; when 
two substances are rubbed together, and one becomes electri- 
fied, electrification of the opposite kind is always developed 
in the other. 

395. Electric attraction and repulsion explai7iable 07i the 
hypothesis of ether-strain. — When small pieces of glass and 
silk are rubbed together, it is found that after they are pulled 
apart they attract each other w^ith a definite and measurable 
force ; and that this force varies inversely as the square of 
the distance between them. When two bodies are pulled apart, 
energy is expended upon them which will be restored when 
they are allowed to approach each other. This phenomenon 
is explainable on the hypothesis that in the work of separa- 
tion, the ether between or around them is strained ; and that 
the tendency of the two bodies to approach each other is the 
tendency of the elastic ether to recover its normal condition. 
The phenomena of electric attraction and repulsion are most 
satisfactorily explained on the hypothesis of ether-strain. By 
whatever hypothesis explained, it is certain that electrification 
is the result of work done, and is a form of potential energy. 

396. What is electricity ? — The student naturally has al- 
ready begun to ask the never-answered question, "What is 
electricity ? " and to inquire, " What is the function of electri- 
city in these operations ? " Provisionally we shall regard elec- 
tricity as that which is transferred from one body to another 
body when the two become oppositely electrified.^ Electricity 

iWhat the ultimate nature of electricity is, whether it he the ether itself or 
(more pi'obably) a constituent of the ether "as water is a constituent of jelly " ; 
whether it he a fluid (it certainly possesses the property of fluidity) ; whether, accord- 
ing to Franklin, a positive charge is an excess and a negative charge a deficit in a 


is 7iot a form of energy.^ It is quite true that electricity 
imder pressure or in inotion possesses energy ; in the same 
sense do water and air under like conditions possess energy, 
but we do not therefore deny them to be forms of matter. 
Electricity, rather, in many respects possesses the nature of 
matter. Like matter it can neither be created nor annihil- 
ated, and like matter it can be moved and put under stress. 
For present purposes, then, electrification may be regarded as 
a state of strain in some intervening medium produced by a 
transfer of electricity from one body to another and the sub- 
sequent separation of the two bodies. Electrification is the 
result of work done, and is most certainly a form of energy. 

397. Quantity of electricity. 

When we do not know what a thing is, it is difficult to conceive 
a definite quantity of it. But our knowledge of electricity like that 
of force is derived from its effects. From the measurement of its 
effects, therefore, can we define a unit quantity of electricity. For 
purposes of calculation at least, electricity of either kind may be 
treated precisely as if it were a material incompressible fluid, and 
any increase or decrease of electrification may be considered to be 
produced by the addition or taking away of a quantity of electricity. 
Quantities of electricity are added and subtracted by the usual rules 
of algebra, the kinds of electrification being denoted by the + and 
— signs. 

398. Latv of attraction. 

When two equally electrified bodies attract or repel each other 
with a force of one dyne at a distance of one centimeter in air, each 
is charged with a certain definite quantity which may be taken as 
a unit quantity of electricity. If two bodies repel or attract each 

certain standard quantity of the fluid which all bodies are supposed to possess in 
their unexcited state ; or whether (more probably) positive and negative electricities 
are distinct entities (whose relations to each other are more like those of sodium 
and chlorine than like those of heat and cold), such that when combined in any 
body they neutralize each other so that the body possesses none of the properties of 
electrification, but when they by any means become separated two separate portions 
of matter always become oppositely charged — are questions too recondite for dis- 
cussion in any general work on physics. Consult " Modern Views of Electricity," 
by O. J. Lodge. 

1 See Daniell's Principles of Physics, p. 530. 



other with some other intensity, the quantities with which they are 
charged are easily determined. For example, suppose that a body 
charged with three units is attracted at a distance of one centimeter 
by one charged with six units. The tol^l attraction of the six units 
of the second body for each one of the other three is obviously 
expressed by six, giving a total attraction of 6 X 3 = 18 dynes. 
It is evident that if any two of these three quantities be known, 
the third can be determined. 

Again, suppose that the attraction (or repulsion) be at some other 
distance, the force, being a radiant one, varies inversely as the square 
of the distance ; consequently, to determine this force the product of 
the two quantities must be divided by the square of the distance 
between them. Now if we substitute quantities of electricity, q and 
g', for masses m and mf in the formula for attraction of gravitation 
(§ 96), we shall have the formula for electrical attraction (or re- 
pulsion) A ; viz. A = z—r- , in which k is the dielectric constant 

(see § 409). That is, the electrical attraction between two charged 
bodies (provided the areas of the bodies are small so as to keep 
them under the law of radiant force) varies as the products of their 
charges and inversely as the square of the distance between them. 

399. Electroscope. — This is an instrument used to detect 
the presence of electrification in a body, and to determine its 
kind. It usually consists of two strips of gold foil, A B (Fig. 

335), suspended from a brass 
rod within a glass jar. To 
the upper end of the rod is 
fixed a metal disk, C. On 
the opposite sides of the in- 
terior of the jar are two strips 
of metal foil, D and E, of suf- 
ficient hight to be touched by 
the strips A and B on their 
extreme divergence. 

(1) If an unelectrified body 

be brought near the disk C, 

Fig. 335. no ciiange takes place in the 


two strips of foil A and B^ but if an electrified body be 
brought near the disk, the strips diverge, thus indicating the 
existence of a charge of electricity in the body. 

(2) If the electroscope be charged by contact with an ex- 
cited body, the strips will remain in a divergent position. 
While in this condition, if a body similarly charged be 
brought near the disk, the strips will diverge more; but if 
an unexcited body, or a body oppositely electrified be brought 
near the disk, the strips will collapse. 

400. Conduction. 

Experiment 6. — a. Eub a brass tube, held in the hand, with warm 
silk. Bring it near the disk of the electroscope ; the leaves are unaffected. 
6. Wrap a piece of sheet rubber around one end of the tube and hold this 
end in the hand, and rub as before. Bring it near the disk of the elec- 
troscope ; notice that the leaves diverge, c. Repeat the last operation ; 
but before bringing the tube near the disk touch the tube with a finger. 
The leaves no longer show signs of electrification. 

In the first (a) and last (c) operations electricity escaped 
through the hand and body to the earth ; in the second (h) it 
was prevented from escaping by the intervening sheet rubber. 
Substances which allow electricity to spread over them, i.e. 
substances which offer little ^resistance to the flow of elec- 
tricity, are called conductors. Those which offer great resist- 
ance to its passage are called non-conductors, insulators, or 

Some of the best insulating substances are dry air, ebonite, 
shellac, resins, glass (free from lead, e.g. common bottle glass), 
silks, and furs. On the other hand, metals are, as a class, 
exceedingly good conductors. Moisture injures the insulation 
of bodies ; hence experiments succeed best on dry, cold days 
of winter, when moisture of the air is least liable to be con- 
densed on the surfaces of apparatus, especially if it he kept 

Water cannot be retained in a reservoir unless its walls be 
of sufficient strength ; so a body, in order to become charged 



and to retain the charge, must be surrounded by something 
that will offer sufficient resistance to the escape of electricity. 
As regards a specific body, there is no limit to the quantity of 
electricity with which it can be charged, provided the charge 
can be retained. This entity which represents the walls of the 
reservoir is termed the dielectviG. It may be the air ^ or any 
of the so-called non-conductors of electricity. Even the ether 
may be considered a dielectric. A body thus surrounded is 
said to be insulated. 

Experiment 7. — Prepare an insulated stool by placing a board on four 
dry and clean glass tumblers, used as legs. Let a person, whom we will 
call John, stand on this stool, and hold in one hand one end of a wire 
(say) 4 yds. long, the other end of which is attached to the disk of an 

_ electroscope. a. Let a 

second person, James, 
strike John with a cat's 
fur ; the leaves diverge. , 
h. Substitute a white silk 
thread for the wire, touch 
the electroscope with a 
finger io as to discharge 
it, and repeat the last op- 
eration ; the leaves do not 
diverge. . c. Let James 
strike John several times 
with the fur, and then 
bring a finger knuckle 
near to some part of 
John's person, e.g. the 
chin, nose, or a knuckle 
(Fig. 336); an electric spark will pass between the two and both will 
experience a slight shock. The electricity which had accumulated on 
John in consequence of his insulation, escapes or is discharged through 

Fig. 336. 

1 If the air were a conductor of electricity, a body could not be charged in it; 
there could be no thunder storms, and man would probably never have known of the 
existence of electricity. 



Section II. 


401. Electricity acts across a dielectric. 

Experiment 1. — Fig. 337 represents an empty egg-shell covered with 
tin foil to make it a good conductor. It is suspended from a glass rod 
by a silk thread, a. Electrify a 
glass rod and bring it near the 
shell. The shell moves toward the 
rod. h. Next introduce a glass 
plate between the rod and shell. 
The shell approaches the rod as 

The chief lesson we learn 
from this experiment is that 
electricity acts across a dielec- 

air : 

In a the dielectric was 

Fig. 337. 

in b, air and glass. 
402. To determine what ac- 
tually happens on an insulated 
conductor when an electrified 
body is brought near. 

Experiment 2. — a. Suspend, as above, two shells so as to touch each 
other, end to end, as in Fig. 338, thus making practically one conductor. 
Bring near to one end of the shells a sealing-wax rod, D, excited with 
— E. While the rod is in this position carry a thin strip of tissue paper, 

C, along the shells. The paper is attracted to the shells, but most 
strongly at the ends. In the middle of the conductor, where the shells 
touch each other, there is little if any electrification. 

6. While the rod D is still in position, separate B from A, then remove 

D. Test each shell with the tissue paper ; both are found to be excited, 
c. Charge an electroscope with +E. Then bring A near it • the leaves 

diverge, showing that A is charged with -HE. Bring B near the electro- 
scope ; the leaves collapse, showing that B is charged with — E. 

1 Insulators across which electric action takes place are called dielectrics, from 
the Greek 5id, across. 



d. Finally bring the two shells near each other ; they attract each 
other. Allow them to touch each other, and then test each with the 
tissue paper or the electroscope ; it will be found that both have become 

From the above operations we learn that when an electri- 
fied body is brought near but not in contact with an insulated 

Fig. 338. 

conductor, the electrified body acts across the dielectric upon 
the conductor, repelling electricity of the same kind to the 
remote side of the conductor, and attracting the opposite kind 
to the side near to it. Such electrical action is called induc- 
tion. The electrified body which produces the action is called 
the inducing body ; the charge of electricity thus produced is 
called induced electricity. 

403. Charging by induction. 

Experiment 3. — Take a proof plane E (Fig. 339) (which consists of an 
insulating handle of glass or gutta percha, terminating at one end with a 
thin metal disk, F, about the size of a 5-cent nickel), and connect it with 
an electroscope, G, by a fine wire, H. Bring a stick of sealing-wax 
electrified as before with — E near the egg-shell conductor. Holding 
the proof plane by the insulating handle, bring the disk near the end of 
the conductor charged by induction with — E. The — E will act induc- 
tively upon the continuous conductor consisting of disk, wire, and 
electroscope, charging the end nearest itself {i.e. the disk) with +E and 



Fig. 339. 

the remote end (i.e. the leaves) with — E. The leaves of the electro- 
scope show the presence of a charge by their divergence. 

Now while everything is in the position indicated by the cut, touch 
with the finger any 
part of the continu- 
ous conductor ; the 
leaves of the electro- 
scope instantly col- 
lapse. The — E with 
which the leaves had 
been charged being 
free is discharged 
through your body. 
But the + E concen- 
trated on the disk of 
the proof plane is 

hound by the attraction of the charge of — E on the end of the shell 
nearest it, and cannot escape. Kemove the finger from the electroscope 
and the proof plane from the influence of the shell; the leaves again diverge. 

The last phenomenon' is explained as follows : After — E 
had been discharged from the continuous conductor, there 
was left an excess of +E; but this excess was all concen- 
trated in the disk F so long as it remained near the negative 
charge of the shell. But as soon as E was removed from the 
influence of the shell, the charge spread itself over the entire 
conductor, and the leaves, which received a portion of the 
charge, diverged. The conductor is said to be charged by 

Ex'periment 4. — To electrify the shell by induction, bring the excited 
wax near it, touch the shell with a finger, remove the finger, and finally 
remove the rod. The proof plane being connected with the electroscope 
and being charged with — E, bring E near to the shell A ; the leaves 
collapse, showing that the shell is charged with +E, which draws the 
— E away from the leaves. 

Observe that when a body becomes charged by induction 
the charge which it receives is opposite in kind to that of the 
inducing body. 


404. Charging by conduction. 

Experiment 5. — Disconnect the proof-plane from the electroscope. 
Charge the electroscope with — E and the shell with +E ; touch the 
shell with the disk of the proof plane, then hold the disk near the 
electroscope ; the divergent leaves collapse, showing that the disk bears 
-f-E which it received by conduction from the shell when they were 
brought in contact. Of course the charge is the same kind as that of the 
body which communicated it. 

405. l7iduction precedes attractio7i. — When a pith ball is 
brought near an electrified glass rod, the -|- E on the rod A 

(Fig. 340) induces — E on the side of the ball B 
nearest A and repels +E to the farther side. 
The -j-E of A and the — E of B therefore attract 
each other ; likewise the +E of A and the +E 
of B repel each other ; but since the former 
charges are nearer each other than the latter 
are, the attraction exceeds the repulsion. 
Fir 340 ^^^* Electrification confined to the outside sur- 

face of a conductor. Metal screens. 

Experiment 6. — Place a tin cup, A (Fig. 341), on a glass tumbler 
coated with shellac and charge it heavily with electricity from an electri- 
cal machine (see Section V.). Introduce a proof- 
plane into the cup and touch the interior surface of 
the cup. Remove the proof-plane and place it near 
the electroscope ; the leaves of the electroscope are 

If a solid metal ball, A (Fig. 342), sus- 
pended by an insulating thread, be electrified 
and then covered with two hemispherical 
metallic cups, B and C, having insulating 
handles, and the cups be afterwards re- 
moved, the ball when tested with the elec- fig.341. 
troscope will be found to have lost all its charge, while the 
cups will be found to be charged. It does not make the 
slightest difference as to the result whether an insulated 



Fig. 342. 

conductor be solid or hollow. Wood covered with tin-foil 
answers the purpose as well as any other body. 

If a hollow conductor he 
charged, however highly, 
with electricity, the whole 
of the charge is found 
upon the outside surface. 
If the electroscope in 
the last experiment were 
placed inside the tin cup, 
or if it be set inside a vessel of wire gauze (e.g. a bird cage), 
and the vessel be charged with electricity or a heavily 
charged body be brought near the vessel, the electroscope will 
be unaffected. This interesting and important fact shows 
that a metallic shell, however thin, entirely screens bodies inside 
it from external electrification,' however great.^ 

407. " Faraday'' s Ice-pail Experiment.''^ 

Experiment 7. — a. Insulate well a tin pail (Eig. 343) and connect it 
with an electroscope. Charge heavily a metal ball suspended by an in- 
sulating thread, with (say) 4-E. Lower 
the ball within the pail ; the pail be- 
comes charged by induction, inside 
with — E and the outside (together with 
the electroscope) with 4-E. The leaves 
of the electroscope diverge. 

h. Touch the outside of the pail with 
a finger ; the free charge of + E escapes 
to the earth (see § 413), and the leaves 
of the electroscope collapse. Remove 
the finger and then withdraw the ball 
slowly from the pail ; the leaves of the 
electroscope slowly diverge, and remain diverged after the ball is removed. 

1 To test this stiU further, Faraday built a cubical cage of 12 ft. edge, of copper 
wire, and lined the interior with paper covered with tin-foil. This chamber was in- 
sulated and put in connection with a powerful electrical machine while working. 
He says : — "I went into the cube and lived in it, using electrometers and all other 
tests of electrical states ; I could not find the least influence upon them, though all 
the time the outside of the cube was powerfully charged, and large sparks and 
brushes were darting from every part of its outer surface." 

Fig. 343. 


c. The ball is now charged with + E and the pail with — E. Bring 
the ball in contact with the pail ; the leaves of the electroscope completely 
collapse, showing that the +E and the — E have combined, neutralizing 
each other and- leaving no excess of either. Hence we conclude that the 
positively charged ball when lowered into the pail must have induced an 
equal charge of — E in the pail. This is generally called the Ice-pail 
Experiment because Faraday in the original experiment used an ice-pail. 

The amount of opposite, electricity induced on surrounding 
conductors hy an electrified body is equal to the hody^s oivn charge. 
One body cannot be charged with a quantity of -|-E without 
an equal charge of — E being established somewhere else, and 
vice versa. The student will bear in mind that whenever in 
his experiments he charges any body with electricity, an 
equal complementary charge always exists distributed over 
neighboring objects or on the walls of the room. When a 
thunder-cloud is charged it has its equal complementary 
charge in the part of the earth nearest it. The sum total 
of all the +E and — E in existence is zero. 

408. Definition of electro-static iyiduction. — We are now in 
a position to understand the following definition : Electro- 
static induction is the action ivhereby a charged body surrounded 
by a dielectric evokes an equal and opposite charge on the inner 
surface of the enclosure containing the body and the dielectric. 

409. Inductive capacity. — The power of transmitting in- 
duction varies with different substances. Across glass, sul- 
phur, and shellac the effect produced by an electrified body is 
different from that across air, the distance being the same. 
The power of a dielectric substance to receive and transmit 
that electric strain which we call induction depends on the 
specific inductive capacity of the substance. 

When we electrify a body, a certain quantity of energy is 
expended, and this is regarded as. the energy of the electric 
charge, and may be recovered by discharging the body. The 
energy is, however, stored in the ether around the body said 
to be charged. 



Section III. 


410. Electric density, — The following experiments by no 
means give exact quantitative measurements, but they are 
suitable for our purpose. 

Experiment 1. — a. Charge a well-insulated metal sphere with +E. 
Touch some point on the surface with a proof-plane and bring it 
in contact with an uncharged electroscope. Notice the amount of 

6. Discharge the proof-plane and electroscope and touch a different 
point on the sphere with the proof-plane, and touch the electroscope as 
before. Notice that there is equal divergence of the leaves. 

Experiment 2. — a. Electrify an insulated pear-shaped conductor (Fig. 

344). Touch the larger end 
of the conductor with a proof- 
plane and bring it in contact 
with an electroscope. Notice 
the amount of divergence of 
the leaves. 

h. Discharge the proof -plane 
and electroscope. Touch the 
pointed end ; notice that on 
bringing the proof-plane in 
contact with the electroscope 
the divergence of the leaves is 
greater than before. In this 

Fig. 344. 

case, therefore, the charge is not uniformly distributed. 

We conclude that the electricity is of eriual density on the 
sphere, but of unequal density on the pear-shaped conductor. 
Electric density is defined as the quantity of electricity on a 
body per unit area. 

It follows, from this definition, that if the surface be 
increased, the quantity of electricity being the same, the 
density is diminished, and vice versa. The relation between 



density and area is readily shown in the following manner : 
Suspend a sheet of tin foil 
from a glass rod (Fig. 345), 
connect the lower end of the 
foil with an electroscope, and 
charge the foil lightly. EoU 
the foil up on the rod, and as 
the surface becomes reduced 
the leaves diverge more widely. 
Distribution of electrifica- 
tion, or the electric density at 

different points on a conductor, depends on its shape. In 
Eig. 346 the distances between the surfaces of the bodies and 
the dotted lines are intended to represent approximately the 
relative densities at different parts of each body. 

ilG. 345. 

FIG. 346. 

411. Effect of points. — As bodies become pointed, the elec: 
trie density increases at the pointed end, until it becomes 
so great that the electricity is discharged. The particles of 
air surrounding the point become heavily charged and are 
repelled ; other particles rush in to take their place, and they 
in turn are electrified and repelled. A current of air pro- 
ceeding from the point, called the " electric wind," is thus 
produced, and the conductor becomes discharged by a process 


somewhat analogous to convection of heat. Pomts or sharp 
edges on a conductor cause a continuous loss of electricity, 
and, therefore, must be carefully avoided in all apparatus 
where they are not essential. 

Section IV. 


412. Electrostatics and electrokinetics. — Electricity may be 
at rest, as in a charged body, or it may be in motion, as in 
the case of a charged body connected by a conductor with the 
earth, when it is discharged through the conductor to the earth. 
It will be shown later on that as long as a floAv of electricity 
continues the conductor along which it flows has properties 
different from those of a simple electrified body. That 
branch of electrical science which treats of the properties 
of simple electrified bodies is called Electrostatics, because 
in them electricity is supposed to be at rest y and that branch 
which treats of electricity in motion is called Electrokinetics. 

413. Eotential. — The fundamental fact of electricity is 
that we are able to place bodies in different electrical con- 
ditions. A charge of electricity, which implies an abnormal 
electrical condition, is the foundation of all electrical phe- 
nomena. We are now to discuss in a very simple manner 
the meaning and use of the very important term potential 
with reference to electricity. 

a. When a charged conductor is connected with the earth, 
a transfer of electricity takes place between the body and 
the earth. 

h. If the body be charged with +E, we say arbitrarily that 
electricity passes to the earth ; but if the body be charged 
with — E, electricity passes /rom the earth to the body. 

c. If two insulated charged conductors be connected with 


each other, electricity may or may not pass from one to the 
other. Now whether electricity passes from one to the other, 
and in what direction it passes, if at all, depends upon the 
so-Gslled potentials of the conductors. 

d. If two bodies have the same potential no transfer of 
electricity takes place between them when they are connected 
by a conductor ; but if the two bodies have different poten- 
tials, there will be a transfer, and the body f7vm which the 
electricity flows is said to be at a higher potential than the 
one to which it flows. 

414. Definition of potential. — The potential of a conductor 
may, therefore, be defined provisionally as the electrical con- 
dition of that conductor which determines the direction of 
the transfer of electricity. 

The term potential is relative, i.e. we compare the potential 
of one body with that of another. 

It is important to have a standard of reference whose po- 
tential is considered to be zero, just as it is convenient in 
stating the elevations and depressions of the earth's surface 
to give the distances above or below sea-level, which is taken 
as the zero of hight. For experimental purposes the earth is 
usually assumed to be at zero potential. A body charged 
with H-E is understood to be one that has a higher potential 
than that of the earth, and a body charged with — E is one 
that has a lower potential than that of the earth. 

415. Analogies. — Potential is analogous, in many respects, 
to (1) temperature, and (2) liquid level. 

(1) When we say that the temperature of air is 20° or 
— 10° C, we mean that its temperature is 20° above or 10° 
below the standard temperature of reference, viz. that of 
melting ice. If two bodies at different temperatures be 
placed in thermal communication, heat will pass from the 
body at a higher temperature to the one at a lower, and will 
continue to do so until both are at the same temperature. 


(2) If two vessels, containing water at different levels, be 
put in communication at their bottoms by a pipe, water will 
flow from the one at a higher level to the one at a lower until 
the Avater is at the same level in both vessels. 

Temperature is not heat ; level is not water ; and potential 
is not electricity, but merely the state of the conductor which 
determines the direction of transfer of electricity. 

All points of a conductor^ when the electricity upon it is at rest, 
are at the same potential, regardless of any difference of density which 
may exist at different points. If it were not so there would be a 
continual flow of electricity from the higher to the lower until 
equilibrium was established, i.e. until all points had the same po- 
tential. We can demonstrate this fact by experiment. 

Experiment 1. — Charge a pear-shaped body, A (Fig. 347), with 
electricity. Connect a proof-plane with an electroscope and touch 
the charged conductor with the 
proof-plane at different points ; 
the leaves diverge just the same 
at all points touched, thus show- 
ing that the potential at aU 
points is the same, although the 
density at different points varies 

(compare this experiment with '^"^ yjg. 347. 

Exp. 2, Section III). 

Observe that it is diffe'rence of potential, or simply potential, and 
not quantity or density, which determines a flow of electricity. 
Water does not flow from a larger or deeper pond into a smaller 
or shallower one unless there is a difference of level. 

416. Electrical capacity. 

If two conductors of the same shape, and surrounded by the 
same dielectric, be charged, it will be found that the larger one re- 
quires a larger charge than the smaller one to electrify it to the 
same potential ; i.e. the larger one has a greater electrical capacity 
than the smaller one. Hence the potential of a conductor depends 
upon its charge and its capacity. If C = the capacity of a con- 
ductor, Q = the quantity or charge of electricity, and V = the 
potential, then ^ 



rrom this we see that the capacity of a conductor is equal to the 
charge necessary to raise its potential from zero to unity. 

The capacities of spheres are found to be proportional to their 
radii. Thus if a sphere charged with 20 units of electricity, and 
having a radius of 4 inches, be brought in contact with an un- 
charged sphere having a radius of 1 inch, and these be afterwards 
separated, the quantity on the large one will be 16 units ; that on 
the small one, 4 units. 

Section V. 


417. Electrophorus. — This apparatus is used to produce 
electrification by induction. It consists of a shallow iron 
dish, A (Fig. 348), filled with sealing-wax. At the center 

of the dish is a protuberance, B, which ex- 
tends just through the wax. A flat brass 
disk, C, has a glass insulating handle. 

Experiment 1. — Strike the surface of the wax 
a few times with a cat's fur, or rub it with a dry 
flannel. The wax becomes electrified with — E. 
Place the disk C upon it. The + E of the disk is 
hound by the — E of the wax, but the — E of the 
disk is repelled by the — E of the wax and passes 
through the protuberance B to the dish below, 
and thence to the earth. Consequently when the 
disk C is raised by the insulating handle from the 
wax, it is charged with +E, and the charge can 
be transferred to any body {e.g. a Leyden jar, see 
§ 420), and then the disk can be recharged by 
replacing it on the wax. This may be repeated 
many times without sensibly decreasing the charge of — E on the wax. 

418. Continuous electropliorus. — Topler-Holtz machine. 

Charging by means of the electrophorus like that described above 
is necessarily intermittent. The Topler-Holtz machine acts as an 
approximately continuous electrophorus, i. e. the act of charging by 
this machine is more nearly continuous. 



Fig. 349 represents this machine in perspective, and Eig. 350 is 
a diagram of its essential parts. On the back of a stationary glass 
plate, F F (Fig. 350), called the field plate, are pasted two quadrants 
of varnished paper, 1 1, called the inductors. In front of the field 
plate is a revolving glass plate upon which are pasted equidistant 
tin-foil carriers, a a a' a' n n\ having a flat metal button on the 
center of each. Two brushes of tinsel, C C, connected with the 
inductors, are so supported as to touch the buttons as they pass, 
and thus a connection is established between the buttons in tem- 
porary contact with the brushes and the inductors. A stationary 

Fig. 349. 

metal rod, A B, has metal combs with pointed teeth attached to it 
near each end. The central teeth of these combs are removed and 
replaced by tinsel brushes. This rod serves as a conductor between 
the two buttons on the same diameter, and may be called the neu- 
tralizing conductor. A second pair of combs, C C, are connected 
with the separable discharging conductor K K'. Connected with 
each part of this conductor is a Leyden jar or condenser. 

The mere contact between unlike substances is sufficient to pro- 
duce a very small incipient charge which, -as the plate revolves, 
rapidly increases to a maximum. In starting this action the two 
parts of the discharging conductor are usually brought in contact. 
As the two parts, K and K', become oppositely charged, they may be 
separated farther and farther apart, and discharges between the two 
extremities, in the form of sparks and brushes, occur at intervals. 



which increase with an increase of distance. By the addition of 
Ley den jars, which also become oppositely charged, the amount of 
charge previous to each discharge is increased, and consequently 
the energy of the discharge and the brilliancy of the spark are 
increased, though the discharges are less frequent. 

As the plate rotates, the two inductors are kept constantly and 
oppositely charged, and as two opposite carriers {n and n' for ex- 
ample) are about to leave the inductors the following takes place : 
At n the positively charged inductor acts through the glass upon the 
carrier and comb, attracting and binding the — E and repelling +E. 
Similar action takes place at n', but with opposite signs. These 

repelled charges unite through the conductor A B and neutralize 
each other, leaving the carriers n and n' charged respectively with 
— E and +E, As n and n' move away from the inductors their 
charges become free, and on reaching the brushes C C they com- 
municate a portion of their charges to the brushes to make good 
any losses by leakage or otherwise which the inductors may sustain. 
We are now able to see how the charges of the inductors are re- 
ceived and maintained. 

We now turn our attention to the discharging conductor. The 
two inductors act inductively upon the two parts K and K' of the 
conductor, charging K with +E and K' with — E. The work re- 


quired to keep up the motion of the revolving plate increases as the 
charges rise, as there is a constant pulling apart, at different points, 
of bodies oppositely charged. Thus mechanical energy becomes 
transformed into electric energy, or the energy of ether strain, i 

The above is a partial description of the action of this machine. 
For a complete description the student may consult larger v^^orks.^ 

419. Condenser. — A very important adjunct to an electrical 
machine is a condenser of some kind, by means of wMcli a large 
quantity of electricity can be collected on a small surface. 

Experiment 2. — Let a person stand on an insulated stool (§ 400), and 
place one hand on the prime conductor of a machine. Let the other 
open hand press agamst a plate of glass or disk of vulcanite, held on the 
open hand of a second person standing on the floor. After a few turns 
of the machine, let the hand that has been on the prime conductor grasp 
the free hand of the second person. Quite a shock will be felt by both. 
Or the connection may be made through a group of persons having hold 
of one another's hands, when the whole company may receive a shock. 

It is evident tliat by this process an unusual quantity of 
electricity had collected previous to the discharge. This 
furnishes an excellent illustration of how electricity may be 
bound by inductive action. The explanation is simple. The 
hand of the first person, charged with + E, acts by induction 
through the glass upon the second person, attracting — E to 
the surface of the glass with which his hand is in contact, 
and repelling -|- E to the earth. Thus, through their mutual 
attraction, the two kinds of electricity become, as it were, 
heaped up opposite ?ach other, and yet are prevented, by the 
insulating glass, from uniting. 

It thus appears that the electrical capacity of a body depends 
not only upon its size (§ 410) but upon tJie presence of charges upon 
other conductors. 

1 After a Topler-Holtz machine has charged a battery of Ley den jars, i.e. stored 
up in the jars electrical energy, the belt may be slipped from the machine (to reduce 
friction), when the battery will drive the machine, reconverting the energy into 
mechanical energy. 

2 Consult Barker, Ganot, Gumming, etc. 



Fig. 351. 

Fig. 352. 

Since C = ^ (§ 416), it is evident that the increase in capacity of 

the conductor is not due to an increase of potential. An electrical 
condenser may be regarded as an appliance for increasing the charge 
without increasing the potential. 

420. Leyden jar. — One of the most convenient forms of 
condenser is the Leyden jar.^ It consists of a glass jar (Fig. 
351) coated with tin-foil both inside and outside to about 
two-thirds its hight. A brass rod passes inside through a 

varnished wooden stopper, and 
touches the inner foil, and 
terminates in a brass knob on 
the outside. 

The jar may be charged by 
connecting one of its coatings 
with the conductor of an elec- 
trical machine and the other 
with the earth. Or it may be charged by connecting the 
outside coating with one of the discharging conductors of 
the Holtz machine, and bringing the other pole near to the 
ball leading from the inner coating. To discharge the jar, 
connect the outer coating with the knob of the jar. To avoid 
a shock in so doing, a discharger is used (Fig. 352), which 
consists of a bent wire terminating at each end with metal 
balls. The wire is held by a glass insulating handle. 

421. Capacity of a condenser. — The edacity of a conden- 
ser is proportional (1) to the area of the metallic conductor ; 

(2) to the specific inductive capacity of the dielectric ; and 

(3) is inversely proportional to the thickness of the dielectric. 

1 So called because one of the first jars was constructed (by Cuneus) at Leyden, 
Holland (1746). The original discovery was, however, made a year earlier by Kleist 
of Pomerania. He happened to touch a charged conductor of an electrical machine 
with a nail protruding from a bottle containing water. On removing the bottle and 
attempting to remove the nail from the bottle he received a violent shock. His 
hand on the outside of the bottle and the water on the inside undoubtedly answered 
the purpose of coatings. 



The low inductive capacity of some kinds of glass renders it 
entirely unsuitable for this purpose. 

To secure greater capacity than a single jar of ordinary 
capacity will afford, several jars, constituting a " battery " of 
jars (Fig. 353), are placed upon a sheet of tin-foil so as to 
connect all the outer coatings, while the inner coatings are 

Fig. 353. 

connected by a wire joining their projecting rods. The sev- 
eral jars are by this means practically converted into one 
large jar.. 

422. Conditmi of the dielectric. Seat of charge. — That 
inductive action is attributable to the dielectric, and not to 
the conductor, is shown by the Leyden jar with movable 
coatings (A, Eig. 354). B is the dielectric ; C is the outer and 
D the inner conductor. The several parts being put together, 
the jar is charged in the usual manner and placed upon an 
insulator. Then the inner conductor, D, is raised by a glass 
rod out of the jar, and afterwards the glass vessel, B, is 
removed from the outer coating. The several parts are now 
tested with an electroscope. The coatings produce little or 
no disturbance of the leaves ; the glass causes a divergence of 



the leaves. On putting the parts together again and dis- 
charging in the usual way, there will be nearly as brilliant a 
spark as if the charged jar had not been dissected. 

This experiment demonstrates that (1) the seat of the 
charges is on the surface of the glass and not on the coat- 
ings ; (2) the coatings serve merely the purpose of conductors 
to spread electricity at the time of charging, and to allow its 
escape from all parts of the electrified surfaces at the time 

Fig. 354. 

of discharge ; (3) a charge is not an electrification of the 
conductors, but, rather, of the dielectric, or, as we shall say 
later on, of the 'Afield" itself, the extent of the conductor 
determining the limits of the field. 

423. Limit of the charge of condensers. — There is a limit 
beyond which a condenser cannot be charged. When elec- 
trification takes place, the stress produced by the opposite 
charges causes a strain in the glass. When the strain be- 
comes too great, a discharge occurs across the dielectric, either 
through the air over the top of the jar, or, if the glass be thin 
enough, by puncturing it. 



Section VI. 


424. Contact action. 

Bring two bodies of dissimilar nature {e.g. a §tick of sealing-wax 
and a woolen cloth) in contact, best by rubbing to secure better 
contact, and separate them, and they exhibit strong attraction for 
each other, ordinarily vastly greater than that of gravitation. The 
contact serves to establish bonds of attraction, i.e. the ether between 
them is supposed to operate like india-rubber bands, pulling the two 
bodies together. The bodies are thus said to be electrically excited. 
To separate the excited bodies requires work to be done ; and the 
bodies when separated possess energy of electrical separation. 

425. Lines of force. Field of force. 

The space or dielectric between and, to a limited extent, around 

the excited bodies is assumed by Faraday to be full of what he 

called lines of force., the positive direction of a line at any point in 

this space being the direction in which a positively electrified particle 

tends to move under the influence of the electrical field. The space 

thus occupied by lines of force is called the field of force. Faraday 
remarks that the stress is as if these lines were stretched 
elastic threads endowecf with the property of shortening 
themselves and also the property of repelling one another 
as well. In other words, there is a tension along these lines 
and a pressure at right angles to them. When bodies 
oppositely excited are brought near together, the lines are 

almost straight from one to the other (except near the edges) of the 

facing areas, as shown in 

Fig. 355. As they are 

more separated the lines 

curve outward, always 

tending to separate from 

one another and from the 

common axis of the two 

bodies, some even curling 

round to the back of the 

bodies, as represented in 

Fig. 356. The expression 

" lines of force " must be Fig. 356. 


Fig. 355, 


regarded as purely a matter of convenience. They have no more 
and no less existence than have "rays of light." 

Section YII. 
atmospheric electricity. 

426. Lightning. — Franklin, by a series of historic experi- 
ments, proved the exact similarity of lightning and thunder 
to the light and crackling of the electric spark. Certain 
clouds which have formed very rapidly are highly charged, 
usually with +E, but sometimes with — E. The, surface of 
the earth and objects thereon immediately beneath the cloud 
are, of course, charged inductively with the opposite kind of 
electricity. The cloud and the earth correspond to the coat- 
ings, and the intervening air to the dielectric, of an immense 
condenser. The opposite charges on the earth and on the 
cloud hold each other prisoners by their mutual attraction. 

As condensation progresses in the cloud its capacity de- 
creases and its potential rise's (since CV^Q). This process 
continues till the difference of potential between the cloud 
and the earth becomes great enough to produce a discharge 
through the air. 

It is the accumulation of induced charges on elevated 
objects, such as buildings, trees, etc., that offers an intensi- 
fied attraction for the opposite electricity of the cloud in 
consequence of their greater proximity, and renders them 
especially liable to be struck by lightning. 

The clouds gather electricity from the atmosphere. Our 
knowledge of the method by which the atmosphere becomes 
charged is very limited. 

427. Lightning-rods. — A good lightning conductor offers a 
peaceful means of communication between the earth and a 
cloud ; it leads the electricity of the earth gently up toward 


the cloud, and allows it to combine with its opposite without 
disturbance, thereby so far discharging the cloud as possibly 
to prevent a lightning stroke ; or, if the stress be too great to 
be thus quietly disposed of, the flash strikes downward, and 
is led harmlessly to the earth by the conductor. A71 ill-con- 
structed lightning-rod may be luorse than 7ione. A good rod 
should be made of good conducting material, so large that it 
will not be melted, and free from loose joints. The lower 
end should be buried in earth that is always moist, and the 
upper end should terminate in several sharp points. Maxwell 
suggests that the best form for a lightning protector is one 
which approximates a net-work covering the entire house 
(see foot-note, p. 445). 

428. The aurora is a luminous phenomenon caused (as 
experiments performed by Lemstrom in Lapland seem to 
indicate) by currents of electricity passing from the higher 
and rarefied regions of the atmosphere to the earth. In the 
Arctic regions the aurora borealis (northern lights) is of 
almost daily occurrence. It sometimes forms an arch, and 
sometimes illuminates the whole sky. 




Section I. 


429. Apparatus required. 

There are required a condensing electroscope, i.e. one whicli has two 
disks separated from each other by a dielectric (Fig. 358), the upper one 
having an insulating handle ; a tumbler | full of water, into which have 
been poured two or three tablespoonfuls of strong sulphuric acid ; a strip 
of sheet-copper, and two pieces of rolled zinc, each about 5 inches long, 
1^ inches wide, and at least ^-^ of an inch thick (a piece of No. 16 copper 
wire 12 inches long should be soldered to one end of each piece of metal, 
and the soldering covered with asphaltum paint) ; 2 yds. of silk insulated 

No. 18 copper wire; two double 
connectors (Fig. 357) which serve 
to join two wires without the in- 
convenience of twisting them to- 
FiG. 357. gether ; and a battery of four vol- 

taic cells (either Bunsen or other 
reputable kind). One of the zincs should be amalgamated as follows : 
First dip the zinc, with the exception of i inch at the soldered end, into 
the acidulated water ; then pour mercury over the wet surface, and 
finally rub the surface, now wet with mercury, with a cloth (to insure 
complete amalgamation, it is best to repeat this operation). 

430. Experiments. 

Experiment 1. — a. Put the unamalgamated zinc into the tumbler 
containing acidulated water. Bubbles of hydrogen gas arise from the 
surface of the immersed zinc. 

6. Remove this zinc and introduce the amalgamated zinc. No bubbles 
(or at least very few) arise from the latter, provided that the zinc is 
properly amalgamated. 



c. Put the copper strip into the liquid, hut do not allow the two metals 
or their wires to touch. No bubbles arise from either metal. Connect 
the wires of the two metals with a double connector ; copious bubbles 
arise from the copper strip, but very few from the zinc strip. Bubbles 
escaping from the copper make it appear as if chemical action were taking 
place between the metal and the liquid. But experience will teach you 
that the appearance is deceptive, as you will find that in no case is 
copper consumed. 

d. Substitute the unamalgamated zinc for the amalgamated ; bubbles 
rise abundantly from the surfaces of both the zinc and copper. 

Lesson learned : An unanialgamated zinc is acted on by the 
liquid under all circumstances ; an amalgamated zinc is not 
acted on by the liquid unless the copper strip is also in the 
liquid, and not then unless the metals are connected. If then 
we would at any time stop the action, we have only to dis- 
connect the metals. It seems also that the wire connecting 
the two metals serves some important purpose in keeping up 
this action. 

Experiment 2. — In this experiment it will be necessary to use metal 
plates of much larger size, or (which will prove much more satisfactory) 
we must use an apparatus somewhat in advance of our present knowledge, 
mz. a battery (§ 4S1) of (say) four 
cells connected in series (§ 482). 

a. Connect the copper to the 
lower disk of. the electroscope (Eig. 
358) by an insulated wire, merely 
touching it with the end of the 
wire, and the zinc to the upper 
disk. Remove the wires, and lift 
the top disk by the insulating han- 
dle. Tlie leaves of the electroscope 
diverge. Prove by suitable test that 
the electrification m the leaves is 

h. Repeat this operation, but 
touch the lower disk with the wire from the zinc, and the upper one with 
the wire from the copper. Show that the leaves have now a negative 

Fig. 358. 


When the upper plate is lifted, the capacity of the condenser 
diminishes considerably, so that the small charge on the lower disk 
raises its potential so much that the gold leaves diverge. This will 


be understood from the equation V = — (§ 416); when the denomi- 

nator C of the fraction — diminishes, the fraction increases, and 
therefore V increases. 

If a plate of metal be placed in a liquid of a class which we 
shall term an electrolyte (i.e. one which is capable of being 
decomposed by a current of electricity), there is a difference 
of electrical condition produced between them so that the 
metal becomes either of higher or lower potential than the 
liquid, according to the nature of the metal and liquid. 

W-e know that if two conductors be at different potentials, 
electricity tends to flow from the one whose potential is 
higher to that whose potential is lower ; if, therefore, two 
dissimilar metals be placed in the same electrolytic liquid 
and we show by actual experiment, as above, that the free 
end of the wire in connection with one plate is charged with 
+ E, and the free end of the other with — E, we conclude that 
if the two oppositely charged bodies be brought in contact, a 
current of electricity will flow from the positively charged 
plate to the negatively charged one. A current therefore 
flows through the connecting wire from the copper (which is 
called the positive electrode) to the wire leading from the zinc 
(which is called the negative electrode), when they are con- 

That difference in quality, in virtue of which zinc and cop- 
per placed in acidulated water can give rise to an electric 
current, is called their electro-chemical difference, and the zinc 
is said to be electro-positive to the copper in the liquid. 
There is a perplexing nomenclature in use by which the zinc 
plate is called the electro-positive element or plate, though it 
is called the negative pole of the combination. 


Section II. 


431. Voltaic cell. — Two electro-clieniically different solids 
(of which zinc is almost invariably one) placed in an electrolytic 
liquid constitute what is called a galvanic or voltaic ^ cell (or 
2)air). One of these plates must be more actively attacked 
by the liquid than the other ; the plate most acted upon is 
called the electro-positive plate, and the other the electro- 
negative one. 

The greater the disparity hetiveen the two solid elements ivith 
reference to the action of the liquid on them, the greater the 
difference in potential ; hence, the greater the current. 

In the following electro-chemical series the substances are so 
arranged that the most electro-positive, or those most affected 
by dilute sulphuric acid, are at the beginning, while those 
most electro-negative, or those least affected by the acid, are 
at the end. Tlie arrow indicates the direction of the current 
through the liquid. 

0^ ^ ri o 
.:1 2 .S <X) O r;^ ^ eg 

It will be seen that zinc and platinum are the two sub- 
stances best adapted to give a strong current. 

When the wires from the two plates are joined the dis- 
charge of the two plates would produce electrical equilibrium, 
were there not some means of maintaining a difference of 
potential between the two plates. This is accomplished by 

1 The original Volta's cell consisted of a plate of copper and a plate of zinc im- 
mersed in dilute sulphuric acid. It is a very inefficient battery, yet with such an 
appliance Sir Humphry Davy performed his classic experiments upon the metals of 
the alkalis, and produced the first voltaic arc. 


the chemical action between the liquid and the electro-positive 
plate and at the expense of the chemical potential energy of 
the electrolyte and plate. A voltaic cell is, therefore, a con- 
trivance which converts chemical energy into electrical energy} 
It should be remembered that it is the role of the battery to 
maintain a difference of potential between the two plates, or 
what is the same thing, between the battery terminals or 
poles, and not to '• generate electricity." 

432. Circuit. — This term is applied to the entire path 
along which electricity flows, and it comprises the battery 
itself and the wire or other conductor connecting the bat- 
tery-plates. Bringing the two extremities of the wire in 
contact and separating them are called, respectively, closing 
and opening, or making and breaking, the circuit. Opening a 
circuit at any point and filling in the gap with an instrument 
of any kind so that the current is obliged to traverse it, is 
called introducing the instrument into the circuit. 

433. Ground circuit, — It was an early discovery in tele- 
graphic history that a complete metallic circuit is not neces- 
sary, but that, in common parlance, the earth can be used as 
a "return circuit." This type of circuit is represented by 
a battery with a wire leading from one plate to any con- 
venient point of the earth, and a second wire leading from 
the other plate to any other point of the earth, which may be 
many miles distant from the first point. No one can assert 
that the current in such a case really goes through the earth 
from one of these points to the other. The earth may be 
regarded as a great resei^voir, rather than as a conductor. 
The battery acts like a pump raising electricity at one end 
from the earth and discharging it at the other end to the 
earth. It is obvious that a pump might be kept in action by 
pumping from an ocean and back into the same ocean with- 
out disturbing its level and yet there would be a continuous 

1 A single voltaic couple is lasually termed a cell ; a combination of cells, a battery. 


flow through the water conductors or pipes. This would 
represent what is known as a ground circuit. 

434, Theory of the electro-chemical action of the battery. — 
The following is a brief statement of Grothus' theory of 
electrolytic action, somewhat modified. The small ovals in 
Figs. 359 and 360 represent molecules of the electrolyte, 
which in this case we suppose to be water,^ each molecule of 
water containing two atoms of hydrogen (Hg) and one atom 

Fig. 359. 

Fig. 360. 

of oxygen (0). The series of molecules a illustrates the 
condition of the molecules before the metals are introduced ; 
the series h represents their condition after the introduction 
of the metals and before the circuit is closed. Eig. 360 rep- 
resents the condition of things at the instant the circuit is 
closed or at the instant of discharge. The molecules are 
polarized and like sides turned in the same direction. At 
this instant there is a redistribution of electricity, an electric 
flow, an equalization of potential. At this instant, chemi- 
cally speaking, there is as it were an interchange of partners, 
the next the zinc combining with it to form ZnO, its Hg 

1 It is, perhaps an open question whether the water (HgO) or the acid (H2SO4) is 
the real electrolyte. For our purpose it does not matter. 


combining with the of the next molecule, and so on through 
the whole row of molecules, until, finally, the Hg appears at 
the surface of the copper, where it is set free. Immediately 
following the discharge and equalization of potential a re- 
polarization (p) takes place followed by a discharge (c). The 
discharges follow one another so rapidly as to furnish prac- 
tically a continuous flow of electricity. 

435. Electrolytic conduction. - — As rapidly as ZnO (zinc 
oxide) is formed it combines with the sulphuric acid and 
forms zinc sulphate. The hydrogen escapes in bubbles, as 
has been shown, from the electro - negative plate. Thus 
oxygen keeps disappearing by combination with the zinc at 
one end of the electrolyte ; the hydrogen, by evolution as a 
gas, disappears at the other. Meantime the molecules keep 
exchanging atoms, so that a constant traveling of the H and 
atoms through the electrolyte is kept up, each conveying 
its own peculiar charge. In this way what is virtually a 
current traverses the liquid. The current, however, does not 
traverse the liquid as it does a wire, but electricity is trans- 
mitted by electrolytic action. The electricity flows not 
through but with the atoms of matter. It is conveyed 
through the liquid by a procession of charged atoms, and the 
process of transmission is more nearly that of convection 
than of conduction. This process is called electrolytic coov- 

Section III. . 


436. Importance of amalgamating the zinc. — All commer- 
cial zinc contains impurities, such as carbon, iron, etc. 
Fig. 361 represents a zinc element having on its surface a 
particle of carbon a, purposely magnified. If such a plate be 
immersed in dilute sulphuric acid, the particles of carbon will 


form with the zinc numerous voltaic circuits, and a transfer 
of electricity along the surface will take place. This coast- 
ing trade, as it were, between the zinc and the impurities on 
its surface, diverts so much from the regular bat- 
tery current, and thereby weakens it. In addi- 
tion to this, it occasions a great waste of materials, 
because, when the regular circuit is broken, this 
local action, as it is called, still continues. If 
pure zinc^ were used, no local action would occur 
at any time, and there would be no consumption of 
material except when the circuit was closed. If 
mercury be rubbed over the surface of the zinc 
after the latter has been dipped into acid to clean 
its surface, the mercury dissolves a portion of the zinc, forming 
with it a semi-liquid amalgam, which covers up its impurities, 
and the amalgamated zinc then comports itself like pure zinc. 
437. Polarization of the negative element. 

Experiment. — Construct a voltaic cell composed of dilute sulphuric 
acid and plates of copper and zinc. Introduce into the circuit a gal- 
vanoscope (§ 447) and note the deflection of the needle when the circuit 
is first closed. Watch the needle for a time. Little by little this deflec- 
tion will decrease, and as it decreases bubbles of gas collect on the copper 
plate. This accumulation of gas is called " polarization ^ of the negative 
element or plate." 

We already understand that a difference of potential is the 
indispensable prerequisite to a flow of electricity. Accom- 
panying a difference of potential there seems to be something 
analogous to a force which is said to cause the flow of, or to 
urge, the electricity through the circuit. The film of gas on 
the copper reduces the electro-chemical difference between it 
and the zinc plate, upon which the generation of this force 
depends, and thereby diminishes the efficiency of the battery. 

1 rormerly pure zinc, obtained at great expense by distillation, was used. 

2 The term polarization in common use is a most senseless term as here applied. 
Polarization in the electrical world is made to cover a multitude of sins. 



Instead of a copper-zinc pair, we soon have a hydrogen-zinc 
pair. A single fluid battery cannot, therefore, yield a current 
of constant strength unless some means is used to remove the 

To overcome this defect some arrangement must be adopted 
to prevent this deposit of hydrogen upon the negative ele- 
ment. The usual method is to employ in addition to the 
dilute sulphuric acid, which we will term the exciting liquid, 
some other substance (usually a liquid) which is a strong 
oxidizing agent, i.e. which can combine with and remove the 
hydrogen as soon as it . is liberated at the negative plate (or 
positive pole). A substance used for this purpose is termed 
a depolarizer. A mixture of a solution of crystals of bi- 
chromate of potassium in water with a suitable quantity of 
dilute sulphuric acid, forms a depolarizer such as is used in 
the so-called hichromate batteries. Other depolarizing sub- 
stances in common use are bi- 
chromate of sodium, nitric acid 
(an excellent depolarizer but 
very objectionable on account 
of the corrosive and unwhole- 
some fumes to which it gives 
rise), chromic acid, peroxide of 
manganese, copper sulphate, 

438. Grenet cell. 

This is a bichromate of potassium 
battery in which two carbon plates, 
C C (Fig. 362), electrically connected, 
and a zinc plate, Z, suspended be- 
tween them by a brass rod, a, are 
immersed in the mixed liquid referred to above. 

This combination furnishes a much more energetic and constant 
current than would be furnished if only dilute sulphuric acid were 
used. But although polarization of the negative element is dimin- 

FlG. 362. 



ished, another detrimental action is substituted. The layer of 
solution next to the carbon plate is soon (say after a constant use 
of half an hour to an hour) deprived of its active oxidizing powers, 
and then polarization of the negative plate and consequent w^eak- 
ening of the current set in. This difficulty may be partially 
remedied by occasionally agitating the liquid or allowing it to rest, 
during which time the various portions of it become homogeneous 
by diffusion, and the liquid near the carbons becomes more active. 

439. Biinsen cell. 

A plan generally adopted to keep the oxidizing liquid away from 
the zinc plate, where it is not wanted and only does harm, is to 
place the carbon plate in an unglazed, porous, earthen cup and to 

Fig. 363. 

Fig. 364. 

surround it with the oxidizing substance. This arrangement, called 
a two-fluid cell, is that adopted by Bunsen (Fig. 363), Grove, Fuller, 
and others. 

In any form of two-fluid cell yet devised, the oxidizing fluid 
sooner or later diffuses through the porous cup and reaches the 
zinc. Destructive action on the zinc then begins, which amalga- 
mation cannot prevent, and a portion of zinc is uselessly consumed 
without effecting anything in the way of generation of electric cur- 
rent. An attempt to prevent this trouble is sometimes made by 
using a solid depolarizer. 



440. Leclanche cell. 

There is a class of galvanic cells in which the negative element 
is protected from polarization by means of metallic oxides. Of 
these the best known is the Leclanche cell (Fig. 364). In this cell 
the carbon plate C is contained in a porous cup P, and packed 
round with fragments of gas-retort coke and manganese peroxide. 
The manganese compound has a strong afiSnity for the hydrogen. 
But the chemical action of solids is sluggish and they quickly 
polarize when in action. They need periodical rest to recover their 
normal condition. Such are called open-circuit batteries, since they 
are suited for work only on lines kept open or disconnected most 
of the time, such as in telephone and bell-ringing circuits. The 
zinc element, Z, which is a rod of zinc, is immersed in a solution 
of ammonium chloride, which is the exciting liquid. 

441. Daniell cell. 

Leaving the hydrogen-generating batteries, we will examine 
briefly another form incapable of this species of polarization. The 
Daniell cell (Fig. 365) uses a solution which, instead of depositing 
hydrogen, deposits copper upon a copper 
negative plate, and hence is free from hy- 
drogen polarization. It contains a copper 
negative and a zinc positive plate. The 
copper plate is immersed in a solution of 
copper sulphate, the zinc in a solution of 
zinc sulphate or dilute sulphuric acid, and 
a porous cup separates the two liquids. By 
the electrolytic action, the zinc combines 
with the sulphuric acid (H2SO4) forming zinc 
sulphate (ZnS04), thereby setting hydrogen 
free. This hydrogen, while on its way to 
the negative element of the copper plate, 
meets the copper sulphate solution (CUSO4) 
which it decomposes, forming sulphuric acid 
again (H2SO4), and setting free the copper, 
which is deposited on the copper plate. 

In some copper sulphate batteries the porous cup is not employed, 
the difference in the specific gravity of the solutions being relied 
upon to keep them separate, as in the so-called gravity cell (Fig. 

Fig. 365. 



366). This form of cell is commonly used on closed circuits 
The chief merits of these cells are 
the complete absence of polari- 
zation of the negative plate (con- 
sequently, the constancy in the 
potential difference of the two 
elements) and the uniformity of 
current which is yielded. 

The kinds of cells that have 
been devised are numberless. The 
voltaic battery will probably long 
continue to be of scientific inter- 
est, but for commercial uses it 
has already become well-nigh ob- 
solete, being replaced by other 
cheaper and more efficient ap- 
pliances for generating electric 
currents, to be hereafter described. 
The battery must, therefore, com- Fig. 366. 

mand in the future relatively less attention in our text-books 
than formerly. 


1. a. What is an amalgamated zinc plate ? 6. A voltaic cell or pair? 
c. An electrode ? 

2. a. How may it be shown that a plate of copper and a plate of zinc 
become electrified when placed in dilute sulphuric acid ? b. Which will 
be positively electrified ? c. If wires leading from these plates be joined, 
what change will occur ? d. What purpose does the connecting wire 
serve ? e. Why ought not the plates of a voltaic cell to touch each other ? 
/. Why ought not the wires of certain voltaic cells to touch each other 
when not in use ? 

3. a. What do you understand by an electric current ? b. What is 
the function of a battery ? c. With what propriety is the zinc plate of. a 
voltaic cell called the positive plate and the negative electrode of a voltaic 
system ? 

4. How does electricity pass from plate to plate within a voltaic cell ? 

5. a. What is meant by local currents ? 6. How may they be pre- 
vented ? 


6. a. Give an example of a counter electromotive force. 6. By what 
other name is it generally known ? c. What harm does it do ? d. How 
may it be prevented ? 

7. If you have the two leading wires of a battery, and the battery be 
concealed, how can you tell which of the wires is connected with the zinc 
plate of the battery ? 

Section TV. 


442. Summary of effects. — The several effects producible 
by an electric current are as follows : — 

(1) Certain compounds in solution can be decomposed by 
causing the current to pass through the solution. This opera- 
tion is called electrolysis. 

(2) A magnetic needle suspended on a vertical pivot will 
be deflected if a wire through which a current is flowing be 
brought near and parallel to it. 

(3) If a wire carrying a current be wound round a rod of 
soft iron, the iron becomes a temporary magnet ; the magnet 
is called an electro-magnet. 

(4) Heat is generated in a wire through which a current 

(5) If the temperature be raised sufficiently at any point 
of the circuit, the conductor at that point becomes luminous, 
as, for example, in the production of the electric light. 

(6) Various physiological effects are produced by a current, 
such as " shocks," a peculiar taste when the poles are applied 
to the tongue, the sensation of light, etc. 

The effects may be classified as electrolytic, magnetic, in- 
cluding (2) and (3) ; heating, including (4) and (5) ; and 



443. Illust7'atwe experiments. (1) Electrolysis. 

Experiment 1. — Take a dilute solution of sulphuric acid (1 part by 
volume to 20), pour some of it into the funnel (Fig. 367), so as to fill the 
U-shaped tube when the stoppers are re- 
moved. Place the stoppers which support 
the platinum electrodes tightly in the tubes. 
Connect with these electrodes the battery 
wires. Instantly bubbles of gas arise from 
both electrodes, accumulating in the upper 
part of the tube and forcing the liquid back 
into the tunnel. Close the passage in the 
rubber tube by turning down the screw of 
the pinch-cock a. Light a splinter of fine 
wood, blow out the flame, leaving it glow- 
ing ; remove the stopper holding the + 
electrode and introduce the glowing splinter 
into the gas in this arm of the tube. It 
relights and burns vigorously, showing that 
the gas is oxygen. Platinum electrodes are 
used, otherwise a portion of the oxygen 
carried to the + electrode would not be set 
free, but would oxidize the metal {e.g. 
copper), instead of appearing as a gas in 
this arm of the tube. Fill this arm of the 
tube with water and stopper it. Invert the 
U-tube ; the gas in the other arm of the 
U-tube collects in the bend of the tube and 
in the small branch tube. Light a match, remove the rubber tube, and 
quickly bring the match near the orifice of the branch tube. The gas 
burns ; it is hydrogen. 

If the experiment be performed with a Hoffman's voltameter'^ 
(Fig. 368), the gas given off at each electrode may be meas- 
ured by the graduations on the arms of the tube. It will 
be found that the volume of hydrogen given off is just double 
that of the oxygen liberated in the same time. There is another 
important use to which this apparatus is put worth men- 

FlG. 367. 

1 Any vessel employed for performing and measuring electrolysis is called a 




Fig. 368. 

tioning at this point, though a little out of place. The 
quantity of gas given off divided by the 
number of seconds the current has been 
flowing is a direct measure of the mean 
strength of the current, i.e. the number 
of units of electricity which flow through 
the circuit in one second. Hence this 
apparatus serves to measure current 

The electrode by which the current en- 
ters the electrolyte is called the anode; 
and that by which the current leaves, 
the cathode. The elements into which the 
electrolyte is broken up are called the ions. 
The ion appearing at the anode is the 
anion; and that at the cathode, the cation. 
444. Relation of the electrodes and battery elements. — It 
will be observed that in both the battery and the voltameter 
the hydrogen is lib- 
erated at the plate 
toward which the 
current is flowing, 
as shown in the 
diagram (Fig. 369). 
Hydrogen and all 
metals appear on 
the plate toward which the current flows, whether in the 
decomposing cell or the battery ; for example, the silver in 
Exp. 4 appears at the cathode, and the copper in the Daniell 
cell (§ 441) is deposited on the electro-negative plate. 

Experiment 2. — Take in a test-tube a quantity of an infusion of purple 
cabbage prepared by steeping its leaves until well cooked. Pour into 
this infusion a few drops of any alkali, such as a solution of caustic soda. 
The infusion is changed thereby from a purple to a green color. In another 


Fig. 369. 



Into this pour a 
The purple is 

test-tube take another portion of the purple infusion. 
few drops of any acid, such as dilute sulphuric acid 
changed to a red. Only acids will turn this infusion 
to a red and only alkalies will turn it to a green. 
Into a rather strong solution of sodium sulphate 
pour enough of the purple infusion to give it a 
decided color. 

Pour some of this colored solution into a V-shaped 
glass tube (Fig. 370). Into each arm of the tube in- 
troduce a platinum electrode and join these to the 
battery wires. Soon the liquid around the cathode 
is turned green, while that around the anode is turned 
red. Evidently, decomposition of the sodium sulphate has taken place. 
An acid and an alkali are the results. 

When a chemical salt is electrolyzed, the base appears at the 
cathode, and the acid at the anode. 

Experiment 3. — Dissolve, by heating, about three grams of pulverized 
potassium iodide in about a tablespoonful of water. Make a paste by 
boiling pulverized starch in water. Take a portion of this paste about 

Fig. 370. 

Fig. 371. 

the size of a pea, and stir it into the solution. 

Wet a piece of writing paper with the liquid 

thus prepared. Spread the wet paper smoothly fig_ 372. 

on a piece of tin, e.g. on the bottom of a tin 

basin (Fig. 371). Press the negative electrode of the battery against an 

uncovered part of the tin. Draw the positive electrode over the paper. 

A mark is produced upon the paper as if the electrode were wet with a 

purple ink. In this case the potassium iodide is decomposed, and the 

iodide combining with the starch forms a purplish blue compound. 

Experiment 4. — Dissolve about 3 g. of silver nitrate in 100 cc of 
water. With this solution nearly fill the electrolysis tank (Fig. 372) 



which accompanies the porte-himiere (§ 323). Place the tank in the 
porte-lumi^re in position to be projected upon a screen in a dark room. 

Connect tlie battery wires with the elec- 
trodes in the tank. A beautiful deposit 
of silver will be made on the cathode, 
spreading therefrom toward the anode, 
and bearing a strong resemblance to vege- 
table growth ; hence it is called the ' ' sil- 
ver tree." In Fig. 373, A represents a 
silver tree deposited from a weak solu- 
tion, and B one from an extremely weak 

445. (2) Magnetic action and 
magnetic field of a straight cur- 
rent. Magnetic lines of force. 

Experiment 5. — Construct a low re- 
sistance battery of (say) four cells. Close 
the circuit and dip the wire into a little 
heap of filings of soft iron. On raising 
the wire you will iind filmgs adhering 

If a wire bearing a very strong current be passed vertically through 
the center of a board on which have been sifted some very fine iron 

Fig. 373. 
in a cluster to the wire (Fig. 

filings, the filings will arrange them- 
selves in circular lines round the 
current-carrying wire (Fig. 375), thus 
furnishing a graphic representation 
of the magnetic field set up by a cur- 
rent. If a small pocket compass be 
carried around and near the wire, the 
needle will at every point take a position tangent to these circular lines 



of filings, whichever way the current passes. If the current be reversed, 
however, the position of the n and s poles of the needle will be reversed. 
This clearly indicates that there is a difference of direction of these 
circular lines according as the current flows in one direction or in the 
other. These circular lines represent the so-called magnetic lines of force 
which occupy a limited space or field round a current-bearing wire. 

446. Deflection of the magnetic needle by a current. 

Experiment 6. — a. Place the apparatus (Fig. 376) so that the mag- 
netic needle, which points (nearly) north and south, shall be parallel with 
the wires Wi and W2. Introduce the + electrode of a battery into 
screw-cup T2, and the — electrode into screw-cup Ti, and pass a current 

Fig. 376. 

through the upper wire. At the instant the circuit is closed the needle 
swings on its axis, and after a few oscillations comes to rest in a position 
which forms an angle with the wire bearing the current. 

b. Break the circuit by removing one of the wires from the screw-cup. 
The needle, under the influence of the magnetic action of the earth, 
returns to its original position. 

c. Eeverse the current by inserting the + electrode of the battery into 
screw-cup Ti, and the — electrode into screw-cup Tg. Again there is a 
deflection of the needle, but the direction of the deflection is reversed ; 
that is, the north-pointing pole (N-pole), which before turned to the 
west, is now deflected toward the east. 

d. Place your right hand above the wire with the palm towards the 
wire, and with the fingers pointing in the same direction as that in which 



the current is flowing, and extend your thumb at right angles to the 
direction of the current (Fig. 377). You observe that your thumb points 
in the same direction as the N-pole of the needle under the current-bearing 

e. Eeverse the current again (so that it will flow northward), place 
your right hand as before (viz. with the palm towards the wire, and with 
the fingers pointing in the same direction as the current) ; your out- 
stretched thumb still points in the same direction as the N-pole of the 

/. Introduce the + electrode of the battery into screw-cup T3 and the 
— electrode into screw-cup T4 so that the current will flow northward 
under the needle. Place the right hand as directed before, except that 
it must be under the wire, so that the wire shall be between the hand 


Fig. 377. — Kight hand above the wire ; 
needle below it. 

Fig. 378. —Bight hand below the wire ; 
needle above it. 

and the needle ; the thumb will point in the same direction as the N-pole 
(Fig. 378). Eeverse the direction of the current in this wire, and apply 
the same test ; the same rule holds. 

The rule for determining the direction of the deflection of 
the IST-poles of a needle when the direction of the current is 
known is this : Place the outsti^etched right hand over or under 
the wire so that the luire shall be 'hetiveen the hand and the 
needle, with the palm towards the needle, the fingers pointing in 
the direction of the current and the thumb extended laterally at 
right angles to the direction of the current; then the extended 
thumb will point in the direction of the deflection of the N-pole. 

Conversely, the direction of the current may be determined 
by ascertaining the direction of the deflection it produces, as 
follows : Place the outstretched right hand over or under the 


wire (always allowing the wire to come between the hand and 
the needle) ivith the palm toivard the needle and the extended 
thumb in the same direction as the N-pole of the needle is 
deflected; then the fingers luill point in the direction the 
curre7it is floiving. 

It will be observed that a deflection is reversed either by 
reversing the current or by changing the relative positions of 
the wire and needle, e.g. by carrying the needle from above 
the wire to a position below it. 

The force exerted by the current upon the needle in deflect- 
ing it is called an electro-magnetic force. 

447. Simple galvanoscope or current detector. 

Experiment 7. — Introduce the + electrode of the battery into screw- 
cup T2 (Fig. 376) and the — electrode into screw-cup T3, so that the 
current will pass above the wire in one direction and below it in the 
opposite direction, as indicated by the arrows. A larger deflection is 
obtained than when the current passes the needle only once. 

If the right-hand test be applied it will be seen that the 
tendency of the current, both when passing the needle in one 
direction above and in the opposite direction below, is to 
produce a deflection in the same direction, and consequently 
the two parts of the current assist each other in producing a 
greater deflection. 

If a more sensitive instrument, i.e. one which will produce 
considerable deflections with weak currents, be required, then 
it will be necessary to pass the current through an insulated 
wire wound many times around the needle, as shown in the 
sectional elevation and plan (Fig. 379). Such an instrument 
is called a galvanoscope or current detector, since one of its 
important uses is to detect the presence of a current. A 
graduated card divided off like that of the mariner's compass 
is placed beneath the needle so that the number of degrees 
of deflection may be read from it. Fig. 380 represents a 



portable detector used by telegraph and telephone line-men 
to detect whether a circuit is complete, to locate faults, etc. 
From the number of degrees of deflection an approximate 


Fig. 380. 

idea of the strength of the current is obtained. The magnetic 
needle is inside the box, that outside being merely an indicator 
attached to the same pivot. 

448. (3) Mag7ietizing effect of an electric current. Electro- 

Experiment 8. — a. Wind an insulated copper wire in the form of a 
spiral round a rod of soft iron (Fig. 381). Pass a current of electricity 
through the spiral, and hold an iron nail near the end of the rod. 
Observe, from its attraction for the nail, that the rod is magnetized. 
A magnet may be provisionally defined as a body which attracts iron. 

h. Break the circuit ; the rod loses its magnetism and the nail drops. 

The iron rod is called a core, the coil of wire a helix, and 
both together are called an electro-magnet. In order to take 
advantage of the attraction of both ends or poles of the 
magnet, the rod is most frequently bent into a U-shape 
(A, Fig. 382). More frequently two iron rods are used, 


connected by a rectangular piece of iron, as a in B of Fig. 

382. The method of winding is such that if the iron core of 

the U-magnet were straightened, or the two 

spools were placed together end to end, one 

would appear as a continuation of the other. 

A piece of soft iron, h, placed across the ends 

and attracted by them is called an armature. 

The piece of iron a is called a yoke. 

449. (4) (5) Heating and luminous effects 

of the electric current. — Construct a low re- 
sistance battery (§ 481) of four to six cells, 

and introduce into the circuit a platinum 

wire, No. 30, about ^ inch long. The wire 

very quickly becomes white hot, i.e. it emits 

white light, which indicates a temperature 

of approximately 1900° C. 

This experiment illustrates the conversion of the energy of 

an electric current into heat energy. In this case the energy 

of the current is said to be 
consumed in overcoming the 
resistance which the con- 

ductor or the circuit offers 
'^^^^^^ to its passage. Heat is de- 
^^^' ^^^' veloped by a current in every 

part of the circuit, because all substances offer some resist- 
ance to a current ; in other words, there are no perfect 
conductors. The small platinum wire offers much greater 
resistance than an equal length of a larger copper wire ; 
whence the greater quantity of heat generated in this part 
of the circuit. All of the energy of any electric current that 
is not consumed in doing other kinds of work is changed into 

Fig. 38.3 represents a calorimeter of simple construction. An 
inverted wide-mouthed bottle has its stopper pierced by two stout 



copper wires, which are united within tlie bottle by a coil of fine 

platinum wire. Through a hole bored in the bottom of the bottle is 

introduced a thermometer, T. The 
bottle contains a known quantity of 
water. Now if a current be passed 
through the coil during a given time, 
it is clear that the amount of heat 
generated can easily be determined 
by multiplying the known mass of 
water by its temperature change, as 
indicated by the thermometer. This 
heat, however, is equivalent to a 
definite quantity of electric energy 
transformed in the wire. 

By means of apparatus of tMs 
kin^ Joule's law was established : 
viz. The number of units of heat gen- 
erated in a conductor varies as (1) its 

resistance, (2) the square of the strength of the current, and (3) the 

time the current flows. 

450. (6) Physiological effects. 

Experiment 9. — Place the copper electrodes of a single voltaic cell on 
each side of the tip of the tongue. A slight stinging (not painful) sensa- 
tion is felt, followed by a peculiar acrid taste. 

Fig. 383. 

Section V. 


451. Importance of electrical vieasicrements. — Less than 
half a century ago the experimental sciences of electricity 
and magnetism were in a great measure collections of iso- 
lated qualitative results. Now, happily, all this has been 
changed. The introduction of the absolute system of units 
has been largely instrumental in changing experimental elec- 
tricity and magnetism into sciences of which the most delicate 
and exact measurement is the very essence. 


The wonderful developments which have been made in 
recent years in electrical science, and which have led to the 
employment of electric energy in connection with a great 
diversity of industrial arts, are almost wholly due to a better 
understanding of what electrical measurements can be made, 
and how to make them. Indeed, little of a liractical nature 
can be done without some acquaintance with the methods of 
making these measurements. 

452. Strength of current. — The expression strength of cur- 
rent means the rate of flow of electricity. The "size" of a 
stream of water, or the " rate of flow " might be indicated by 
stating the number of gallons which flow past a given point 
in a minute. We have not adopted in hydraulics any par- 
ticular name for a gallon a. minute, but there is a necessity in 
electricity for a term to denote the corresponding idea ; in 
other words, there is a necessity for a unit for measuring 
rate of flow or current strength. 

The so-called C.G.S. electro-magnetic unit^ of current strengtli 
is determined as follows : (First it is necessary to define a mag- 
netic pole of unit strength. If a long thin magnet be broken in 
the middle, the broken ends develop two opposite poles of equal 
strength. Suppose these poles to be placed one centimeter apart 
and the attraction between them to be measured by a very delicate 
spring balance. If these poles are of such strength that at a dis- 
tance of one centimeter they attract each other with a force of one 
dyne, they are said to be of unit strength.) 

Suppose that a thin wire is bent into a circle of 1 cm radius (Fig. 
384), and a magnetic pole of unit strength is held at its center. As 
we shall see further on, there is a force tending to move the mag- 
netic pole along the lines of force of the magnetic field developed 
by the current, i.e. in a direction at right angles to the plane of 
the cncle. Let this force be measured in dynes, which can be 

iThe units which we are here discussing are called electro-magnetic units to 
distinguish them from units of a different nature called electrostatic units (see 
Section VI), which are derived from the effects of electrostatic attraction and repul- 
sion. The electrostatic units are chiefly used in connection with electrostatics, and 
the electro-magnetic units in connection with electrodynamics. Of the electro- 
magnetic units there are two systems, the C.G.S. and the practical. 



done by weighing in grams and multiplying the number of grams 
by 980 (the value of g). Then since the length of the circular 
current is 2 7rcm, the attraction of a unit length 
2 of it is obtained by dividing the total force in 

dynes by 2 7r. If now there be passed through 
the wire a current of such a strength that this 
force per unit of length is equal to one dyne, the 
current is called a unit current. 

A C.G.S. electro-magnetic unit of cur- 
rent strength is defined to be the strength 
of a current such that a centimeter of it acts 
on a unit magnetic pole with a force of one 
dyne, every point of the current being at a 
distance of one centinfieter from, the pole. 

We have found in dynamics that, in 
practical work, the C.G.S. units are seldom 
used on account of their inconvenient size. 
The same applies to electricity. For prac- 
tical use there was adopted by a Congress 
of Electricians a class of units called the practical units, which 
are certain multiples or submultiples ^ of the C.G-.S. units. 

The practical unit of current strength, called the ampere, is 
one-tenth of the C.G.S. unit. The quantity of electricity 
conveyed per second by a current whose strength is one 
ampere is called one coulomb. The coulomb is the practical 
unit of quantity of electricity. 

453. Electro-motive force. The volt. — Water flows from 
one place to another in virtue of a difference of pressure be- 
tween the two places, and the flow takes place from the place 
of high pressure to the place of low pressure. For instance, 
when water flows from a reservoir or cistern the pressure at 
any point in the pipe is due to the " head " of water above it. 

Fig. 384. 

1 These practical units are derived from the C.G.S. electro-magnetic ones by- 
adopting a new unit of length, the earth's quadrant (lO^ cm), and a new unit of 
mass, 10—11 g, the second being retained as a unit of time. 


If it be set flowing by a force pump, we might say the flow of 
water was due to a water-motive force which could be ex- 
pressed as equal to a " head " of a certain number of feet of 

Similarly, electricity flows in a conductor only when there 
is a difference of what may be termed electrical lyressure be- 
tween its ends. If such be maintained between two points 
connected by a conductor, it obviously represents a kind of 
curvent-prodiicing force, one which can keep electricity in 
motion against resistance. It is for this reason called electro- 
motive force (E.M.F.). Electro-motive force is that which 
maintains or tends to maintain a current of electricity through 
a conductor. That which hinders the current is called re- 

Difference in electrical pressure we have hitherto assumed 
to be due to difference of potential. It is this difference of 
electrical pressure which sets up a current in the conductor. 
Potential difference may be due to contact of dissimilar sub- 
stances, as in the voltaic cell, or to the movement of a part 
of the conductor in a magnetic field, as in the dynamo (§ 525). 

Experience shows that if electricity be made to move in oppo- 
sition to E.M.F., or, to speak figuratively, be carried up hill, 
such a displacement of electricity against electrical stress requires 
an expenditure of energy ; that is, it cannot be done without doing 
work and drawing upon a supply of energy of some form. Dif- 
ference of potential is therefore measured by the work done in con- 
veying a unit of electricity in opposition to E.M.F. The C.G.S. 
unit difference of potential is said to exist between two points where 
one erg of work must be done in conveying a C.G.S. unit of elec- 
tricity from one point to the other. 

The volt is the name chosen for the practical unit of E.M.F. 
and difference of potential, and this unit is equal to 10^ 
C.G.S. units. For purposes where great accuracy is not 
required, it will answer to consider a volt as the E.M.F. of 
a Daniell's cell ; i.e. it is aboict the difference of potential be- 


tween the zinc and copper of this cell, the E.M.F. of a 
standard Daniell cell being approximately 1.07 volts. 

454. Electrical luork and electrical activity. The joule OAid 
watt. — The volt is of snch a magnitude that one conlomb of 
electricity conveyed against an E.M.F. of one volt requires 
an expenditure of one joule = one volt-coulomh of energy. The 
volt-coulomb is analogous to the foot-pound and kilogram- 
meter. Hence if a coulomb of electricity flow between two 
points in a conductor whose difference of potential is one 
volt, then one joule of work is done thereby. 

If a conductor be traversed by a current of one ampere, i.e. 
a coulomb per second, and we find two points whose difference 
of electrical level is one volt, then the rate at which work is 
being done in that portion is one tvatt = one joule per second. 
The joule and watt are units of electrical work and electrical 
activity, respectively. 

455. Resista7ice. The ohm. 

In every case in which a steady electric current flows in a con- 
ductor, it is found that if tlie difference in potential between any 
two points in the conductor be measured in volts, and this number 
be divided by the strength of the current in amperes, a quotient 
is obtained which has a constant value. That is, if the difference 
of potential be doubled, the flow will be doubled, and so on. This 
constant ratio of electric pressure to electric flow is called the 
" electrical resistance of the conductor ; that is, 

resistance = pressure -i- flow. 

If pressure be measured in volts, and flow in amperes, the resist- 
ance of a conductor in which a difference of pressure of one volt 
produces a flow of one ampere is called one ohiii^^ and we have 

1 Summary of Practical Electrical Units. 
Names embalmed in scientific nomenclature. 
English. French. German. Italian. American. 

James Watt. Charles A. Coulomb. G. S. Ohm. A. Volta. J. Henry. 

James P. Joule. Andre M. Ampere. 

Names of units called after the above. 
Poioer, the watt. Quantity, the coulomb. Resistance, the ohm. Self-induction, 
Work, the joule. Cxirrent, the ampere. Pressure, the volt. the henry. 


pressure in volts -!- flow in amperes = resistance in ohms ; 

or, divide the difference in potential between any two points by the 
strength of the current, and the quotient is the resistance between 
those two points in ohms. Hence resistance may be defined as the 
ratio of the E.M.F. to the current strength. 

The unit of resistance is called the ohm. Every substance 
offers resistance to the passage of a current. Those sub- 
stances which offer an immensely powerful barrier are called 
insulators. Yet all substances conduct to some extent ; and 
when an insulator is spoken of the term is only relative. 

The ohm,^ as defined by the Paris Congress of Electricians 
(1884), is the resistance ojfered by a column of pure mercury at 
0° C, 1 sqiuvre millmieter in section and 106 centimeters longp- 
It is about the resistance of 9.3 feet of No. 30 (American 
gauge) copper wire (.01 in. diam.) at ordinary temperature. 

The dimensional equation for quantity of electricity is 


1 A megohm is one million ohms. A microhm is one millionth of an ohm. 

2 In July, 1894, an act of U*. S. Congress Avas passed " To define and establish the 
units of electrical measure." The following are quotations from this act. " The 
unit of resistance, known as the international ohm, shall be represented by the 
resistance offered to an unvarying electric current by a column of mercury at the 
temperature of melting ice, fourteen and four thousand five hundred and twenty-one 
ten thousandths grams in mass, of a constant cross-sectional area, and of the length 
of one hundred and six and three-tenths centimeters." 

" The unit of current shall be known as the international ampere and is the 
practical equivalent of the unvarying current, which, when passed through a solu- 
tion of nitrate of silver in water in accordance with standard specifications, deposits 
silver at the rate of one thousand one hundred and eighteen millionths of a gram 
per second." 

The international volt "Is the electro-motive force that, steadily applied to a 
conductor whose resistance is one international ohm, will produce a current of an 
international ampere." 

The international coulomb " Is the qiiantity of electricity transferred by a cur- 
rent of one international ampere in one second." 

The international " Unit of work shall be the Joule, which is practically equiv- 
alent to the energy expended in one second by an international ampere in an inter- 
national ohm." 

The international unit of activity " Shall be the Watt, which is practically equiv- 
alent to the work done at the rate of one Joule per second." 


The dimensional equation for current strength is 

(C) = [M^L^T-i]. 
The dimensional equation for electric potential or E.M.F. is 

(V) = [M^L*T-2]. 
The dimensional equation for resistance is 
(R) - [LT-i]. 

456. OJim^s Laiv. — The three factors, current (C), electro- 
motive force (E), and resistance (E), are evidently inter- 
dependent. Their relations to one another are stated in the 
well-known Ohm's Law thus : The current is equal to the 
electro-Tnotive force divided hy the resistance ; or 

Hence the strength of a current is directly proportional to 
the E.M.F. and inversely proportional to the resistance. 
This famous law is at the base of a large portion of electrical 
measurements, and its applications are developed in treatises 
on the mathematics of the subject. 

457. Resume. — The ampere is analogous to the "miner's 
inch" used by miners and irrigators in the Western States. 
The latter denotes the rate of flow of water which, under a 
head of six inches, will pass through a hole one inch square 
in a board two inches thick. Let this head of water represent 
a volt, and the resistance of the hole one ohm; then the 
miner's inch would represent a current of one ampere. The 
expression "one miner's inch per second " is redundant; so 
is the expression " one ampere per second." 

A unit current is a current maintained by a unit E.M.F. 
against a unit resistance. 

A unit E.M.F. is the E.M.F. required to maintain a unit 
current against a unit resistance. A conductor has a unit 
resistance when a unit E.M.F. or a unit difference of poten- 


to pass 

tial between its two 
through it. 



a unit curreni 

The ampere 
The volt 
The ohm 

= 10-1 

= 108 
= 109 


units of current. 
" E.M.F. 
" resistance. 

From the numerical values given above it will be seen that if we 
have a circuit in which the resistance is one ohm and the E.M.F. 
one volt, then the strength of the current will be one ampere ; for 


^^ = 10-1 


Section VI. 


Eeference has been made in the foregoing pages to various elec- 
trical quantities, and we give below definitions of certain electro- 
static units, rather as a convenience to the student for reference 
than for practical use in connection with this work. 

The electrostatic units embrace the units of quantity, potential, 
and capacity. No names have yet been adopted for these units. 

458. Unit of quantity. 

One absolute unit of electricity is that charge on a very small 
body which, if placed at a distance of one centimeter from an equal 
charge, will exert through air a force of one dyne. 

We can express the force between two quantities q and q' con- 
densed in points d cm apart by —- (Compare with the law of 
gravitation, §96.) The dimensional is g = [M*L^T-i]. 

459. Unit difference of potential. 

Since potential represents work done on a unit of electricity, a 
unit difference of potential may be defined as such a difference 
of potential between two points as requires the expenditure of one 
erg of work to transfer a + unit of electricity from one point to 
the other, that point being at higher potential to which the + unit is 
carried. The dimensional of difference of potential is d = M^L^T-i. 


460. Unit of capacity. 

Since capacity is quantity per unit potential, a unit of capacity 
is defined as such a capacity of a conductor as requires a charge of 
one unit of electricity to raise it to unit of potential. Capacity, 

c, is measured by ^, and its dimensional is c = L. 

461. Electric force and intensity of electric field at a poi7it. 

The electric force at any point in an electric field or the intensity 
of an electric field at any point is the force with which a unit of 
+ E would be acted on if placed at that point. Its dimensional, /, 

is / = [M L T-2] -f [M* L"'" T-i] , or [M* L"* T-i] . 

462. Specific inductive capacity. 

The specific inductive capacity of a dielectric is the ratio of the 
capacity of a condenser, the space between the plates of which is 
filled with the dielectric, to the capacity of a precisely similar con- 
denser with air as a dielectric. It is therefore simply a numerical 

Section VII. 


463. Activity of a current. — The unit of electric activity 
(or rate of doing work) is the ivatt. A watt is the activity of 
a current of one ampere maintained by a difference of poten- 
tial of one volt. 1 volt X 1 ampere =^ 1 volt-ampere or watt. 

(1). A (watts) = C (amperes) X E (volts). 

(2). The watt =c= (10- ^ X 10'^ =) 10^ ergs per second, or 

— — horse-power. Hence = activity in horse-power. 

For example, to find the rate at which energy is trans- 
formed in an electric lamp, measure the whole current in am- 
peres ; measure the difference of potential (with a voltmeter) 
between the terminals of the lamp, in volts ; multiply together 


the quantities thus obtained and divide by 746 ; the result 
will be the horse-power absorbed in the lamp. That is, a 
current of C amperes falling E volts will perform, in passing 

through the instrument, work at the rate of horse-power. 


a. Substituting in equation (1) the value of C as given in 

E E^ 

Ohm's formula, C= — , we have, A=— ; i.e. the activity is 

equal to the square of the E.M.F. divided by the resistance. 

&. E = CR (Ohm's formula). If this value of E be substi- 
tuted, formula (1) becomes A := C^E ; i.e. the activity is di- 
rectly x^'i'oportioiial both to the square of the current strength 
when R is constant and to the resistance when C is constant. 

(3). The amount of chemical decompositio7i produced by a 
current in a given time varies as the strength of the current. 
On this principle is constructed the voltameter, which measures 
the strength of a current by the amount of chemical action it 
effects in a given time. 

(4). The mass in grams of an element deposited by elec- 
trolysis is found by multiplying its electro-chemical equivalent 
(i.e. the mass in grams of the element deposited by one am- 
pere in one second) by the strength of the current in amperes, 
and this product by the time in seconds during which the cur- 
rent electrolyzes. 

(5). The number of units of heat developed in a conductor is 
pjroportional (1) to its resistance, (2) to the square of the strength 
of the current, and (3) to the time the current is flowing. 

A current of one ampere flowing through a resistance of 
one ohm develops therein 0.00024 calorie of heat per second. 
Hence H (calories) = C^ (amperes) X R (ohms) X t (seconds) 
X 0.00024. 

Whenever the current heats a wire, produces decomposition, 
or performs work of any kind, each of these acts is accom- 
plished at the expense of the potential energy in the battery. 


If the current operate an electric motor which pumps water, 
or lifts a hammer, the battery loses energy proportional to the 
work required for each of these mechanical acts. 

Section VIII. 


Our attention is next directed to the consideration of in- 
struments for measuring the quantities which, we have seen, 
are required to be known. First we consider the instrument 
for measuring quoMtity of electricity, properly called a coulomb- 

464. Coulomh-m&ter. Voltameter. — The simplest quantity- 
meter is based on the electrolytic effect of the current, and is 
called a voltameter. If, for instance, using the Hoffman vol- 
tameter (Fig. 368), we measure the hydrogen generated during 
a given time, the mass or volume of this hydrogen under con- 
stant pressure and temperature is exactly proportional to the 
number of coulombs of electricity which have passed through 
the liquid. The mass in grams of any constituent of an elec- 
trolyte liberated by the passage of one coulomb of electricity, 
is called its electro-chemieal equivalent. For commercial pur- 
poses, instead of the coulomb as the unit of quantity, a larger 
unit, the amxjere-hour (^3600 coulombs), is frequently used. 

The following are the electro-chemical equivalents ^ per 
coulomb and per ampere-hour for a few metals : — 

Electro-chemical Equivalent per 

equivalent per coulomb. ampere-hour. 

Hydrogen ; . . .000010354 grams .03727 grams. 

Silver 00111800 " 4.0248 

Copper 0003284 " 1.1822 

Zinc 00033696 " 1.223056 . " 

Lead 00107160 " 3.85776 

1 Tlie student in chemistry will understand that the electro-chemical equivalents 
of different metallic elements are proportional to their combining equivalents. 


Edison in his system of electric lighting employs a zinc 
voltameter for measuring the quantity of electricity furnished 
to each customer. It consists of two zinc plates immersed in 
a solution of zinc sulphate, and is so arranged by means of a 
divided circuit (§ 477) that only a portion (say a thousandth) 
of the current passes through the liquid. The increase of 
mass of the electro-negative plate in grams divided by 1.223 
gives the quantity in ampere-hours which has passed through 
the voltameter. 

In like manner the electrolytical action in the voltaic cell 
itself is proportional to the strength of the current while it 
passes. One coulomb of electricity in passing through a 
Daniell's cell dissolves .00033696 gram of zinc and deposits 
.0003284 gram of copper. 

A coulomb-meter will serve as an ampere-meter (abbreviated, 
ammetery when the current is very nearly constant, but not 
otherwise. Eor the quantity of electricity measured in 
coulombs which has passed through the circuit in a given 
time, divided by the number of seconds, obviously gives the 
coulombs-per-second, or the mean ampere strength of the 

465. Galvanometer. — This is an instrument for measuring 
current-strength by means of the deflection of a magnetic 
needle when placed in the field of the current. It is so con- 
structed that either the deflection angle itself, or some 
function of it, is proportional to the current-strength. 

466. Thompson's mirror galvanoTneter. — A simplified form 
is shown in Fig. 385, and the complete instrument is shown 
in Mg. 386. Insulated wire is wound on a bobbin, A. With- 
in this bobbin is hung, by a silk fiber, a little circular concave 
mirror, to the back of which are attached little magnets of 
watch-spring steel. To adjust it for use, the suspended 
magnets must be set parallel to the coil of wire. For this 
purpose it is necessary to have a small controlling magnet, 



n s, to cause the needles to take the required position. The 
galvanometer is also rendered more or less sensitive by moving 
the controlling magnet farther from or nearer to the needles. 

Fig. 385. 

If a beam of light from a lamp in a dark room be thrown 
upon the mirror and reflected thence to a screen, S, a spot of 

Fig. 386. 

light thereon will show the slightest change in the position 
of the mirror-needle in the coil. Hence a very feeble current 



declares its presence by causing the spot of iiglit to move to 
one side or the other of a central or zero point where the spot 
falls when there is no current. With this galvanometer no 
appreciable error is committed in considering the current 
strengths as proportional to the scale-readings. This instru- 
ment is of great value to the electrician in dealing with very 
weak currents. 

467. Tangent galvanometer. ■ — ■ A tangent galvanometer is 
one so constructed that the current passing through it is pro- 
portional to the tangent of the angle of deflection produced. 
To this end it is necessary that the needle be very short (not 
more than gL) in comparison with the diameter of the coil. 

In its simplest form it consists of a large vertical coil 
(better two coils, one on each side of the needle, Fig. 387, 
so placed that the needle is 
at the center of the common 
axis), in the center of which 
is either a small compass 
needle or a needle suspended 
by a silk fiber. 

A needle thus placed in the 
field of a current is acted on 
by a mechanical couple tend- 
ing to place it at right angles 
to the plane of the coil, and it 
is deflected until this couple is 
balanced by the return couple 
due to the earth's magnetism. 
The value of the earth's mag- ^'''- ^^^• 

netic intensity in a horizontal plane is denoted by H. 

The formula for the tangent galvanometer is C = 


2 7rn 


Ktan 6, in which C is the current strength in C.G.S. units ; H, as 
above, measured in dynes ; 6, the angle of deflection ; and r, the 


mean radius of the coil of n turns. K, which is called the reduction 
factor of the galvanometer, is usually written in the form 

2wn G 
It is made up of two patts, viz. H, which is dependent on locality, 
and G, which depends on the construction of the instrument, and is 
therefore called the galvanometer constant. Hence, if the value of 
H and G be once found, the strength of any current is calculated 
by multiplying the tangent of the deflection angle by the ratio K. 
As an ampere = 10— i C.G.S. unit, the current strength in amperes 
is found by multiplying this value by 10. 

When the scale is divided into degrees, the corresponding tan- 
gents are found by consulting a table of tangents (p. 626). In some 
instruments, however, the scale is graduated directly in tangents. 

The process of finding the value of the reduction factor of any 
instrument is called standardizing. There are many ways of doing 
this, which may be found in any good laboratory manual. One of 
these consists in introducing a copper voltameter into circuit with 
the galvanometer. After passing a current for a certain time, and 
observing the deflection of the galvanometer during that time, ascer- 
tain the gain of mass in grams of the negative plate, and divide this 
gain by the time in seconds. This gives the deposit of copper per 
second. Dividing this result by .0003284 (§ 464) gives the ampere 
current which passed through the galvanometer. Finally, divide 
the last result by 10 tan 6 and the result is the value of K. That 
is, the reduction factor of a galvanometer = strength of current in 
amperes -r 10 tan 6, when 6 is the average deflection, i 

It has been found that errors of observation affect the value of C 
least when the mean deflection is 45° ; hence, it is customary to 
arrange the experiments so that about this deflection angle may 
be produced. 

If the strengths of two currents are to be compared, it is 
only necessary to obtain deflections with each current sepa- 
rately, and compare the tangents of the angles. 

468. Ammeter. — Now if the value of each division of the 

1 It is to be borne in mind tliat the formula C = K tan measures the current in 
C.G.S. electro-magnetic units and not in amperes ; and that 1 C.G.S. unit is equal to 
10 amperes. 



scale be found by multiplying the number indicating the 
tangent of the angle of deflection by 10 K, and these results 
be placed iipon the scale in place of degree numbers, we shall 
have a direct-reading ampere-meter (ammeter). There is a 
great variety of ammeters in use, for a description of which 
the student is referred either to technical works on the sub- 
ject or to the inventors themselves. 

We shall consider only one other form, that called the 
(Kohlrausch) solenoid ammeter, selecting this because of its 
simplicity. In Eig. 388, a is a helix of thick wire, ^'is a soft 
iron tube which serves as a core, suspended by a light spring 

Fig. 388. 

c. The core carries a marker ^y / is merely a wooden guide- 
rod for the tubular core. The action of this ammeter depends 
upon the principle that when an insulated wire is wound into 
a helix (called also a solenoid), and a current is passed 
through the wire, an iron rod or tube placed at the opening 
will be drawn into the helix with a force increasing with the 


strength of the current. This force acting against the elastic 
force of the spring may be measured in the same manner as 
weight by a spring balance. Now as the current strength 
bears a definite relation to the force, the instrument can 
easily be calibrated in amperes. 

Section IX. 


469. External and internal resistance. — For convenience 
the resistance of an electric circuit is divided into two parts, 
the external and the internal. External resistance includes 
all the resistance of a circuit except that of the generator, 
while the latter is termed internal resistance. 

When the external resistance in a circuit is considered 
separately from the internal, Ohm's formula must be con- 
verted thus (calling the former R, and the latter r): — 

If a cell have E = 1 volt, and r = 1 ohm, and the connecting 
wire be short and stout, so that E may be disregarded, then 
the cell yields a current of one ampere. If by any means 
the internal resistance of this cell can be decreased one-half, 
it will then be capable of yielding a two-ampere current 
under the same conditions. 

470. External resistance. 

Experiment 1. — Introduce into a circuit a galvanometer, i and note 
the number of degrees the needle is deflected. Then introduce into the 

1 The galvanometer represented in the cut is a form of galvanometer chiefly used 
by the author in elementary laboratory work. The results obtained by it are ap- 
proximately those which would be obtained by a standard tangent galvanometer. 
The manifold uses to which galvanometers are put in a physical laboratory properly 
require a variety of instruments, and this would make a complete equipment qxiite 



same circuit the wire on tlie spool numbered 4 on tlie platform, 2 S 
(Fig. 389). (The wire on any one of the five spools on this platform can 
at any time be introduced into a circuit, by connecting the battery wires 
with the binding screws on each side of the spool to be introduced.) 

Pig. 389. 

The deflection is now less than before. The copper wire on this spool 
is 16 yards in length ; its size is No. 30 of the Brown and Sharpe wire 
gauge. When this spool is in circuit, the circuit is 16 yards longer than 
when the spool is out. The effect of lengthening the circuit is to weaken 
the current, as shown by the diminished deflection. 

Experiment 2. — Next, substitute Spool 2 for Spool 4. This contains 
32 yards of the same kind of wire as that on Spool 4, The deflection is 
still smaller. 

The weakening of the current by introducing these wires is caused by 
the resistance which the wires offer to the current, much as the friction 
between water and the interior of a pipe impedes, to some extent, the 
flow of water through it. The longer the pipe the greater is the re- 
sistance to the flow. 

If the wire on the spools had been the only resistance in the circuit, 
then, when Spool 2 was in the circuit, the resistance would have been 
double what it was when Spool 4 was in the circuit, and the current, 
with double the resistance, would have been half as strong. 

2 The platform of spools containing wire of different (known) sizes, lengths, and 
material, so arranged that any one, two, or more can he introduced into the circuit 
for the purpose of measurement of resistance, is an instrument of great convenience 
in a school laboratory. 


(1) Other things being equal, the resistance of a conductor 
varies as its length. 

Experiment 3. — Next substitute Spool 1 for Spool 2. This spool 
contains 32 yards of No. 23 copper wire, — a thicker wire than that on 
Spool 2, but the length of the wire is the same. The deflection is now 
greater than it was when Spool 2 was in circuit. This indicates that the 
larger wire offers less resistance. 

Careful experiments show that (2) the resistance of all 
conductors varies inversely as the areas of their cross sections. 
If the conductors he cylindrical it varies inversely as the square 
of their diameters. 

Experiment 4. — Substitute Spool 5 for Spool 1 , and compare the 
deflection with that obtained when Spool 4 was in the circuit. The de- 
flection is smaller than when Spool 4 was in circuit. The wire on these 
two spools is of the same length and size, but the wire of Spool 5 is Ger- 
man-silver. It thus appears that German-silver offers more resistance 
than copper. , 

(3) In obtaining the resistance of a conductor, the specific 
resistance of the substance must enter into the calcidation. 
(See table of specific resistances in the Appendix.) 

The resistance of metal conductors increases slowly with 
the temperature of the conductor? The resistance of German- 
silver is affected less by changes of temperature than that of 
most metals ; hence its general use in standards of resistance. 

471. Internal resistance. 

Experiment 5. — Connect with the galvanometer the copper and zinc 
strips used in Experiment 1, Section 1, and introduce the strips into a 
tumbler nearly full of acidulated water. Note the deflection. Then raise 
the strips, keeping them the same distance apart, so that less and less of 
the strips will be submerged. As the strips are raised, the deflection be- 
comes smaller. This is caused by the increase of resistance in the liquid 
part of the circuit, as the cross section of the liquid lying between the 
two strips becomes smaller. 



(4) The internal resistance of a circuit, other things being 
equal, varies inversely as the area of the cross section of the 
liquid between the two elements. 

In a large cell the area of the cross section of the liquid 
between the elements is larger than in a small cell, con- 
sequently the internal resistance is less. This is the only 
way in which the size of the cell affects the current. 

Obviously the resistance of the battery would be increased 
by any increase of the distance between the elements, since 
this increases the length of the liquid conductor, but as this 
distance is usually made as small as convenient, and is kept 
invariable, it demands little of our attention. 

Section X. 


472. Description of the resistance box. 

Fig. 390 represents a wooden box containing what is equivalent to a 
series of coils of German-silver wire, whose resistance ranges from 0. 1 ohm 
to 100 ohms.i Each of 
these coils is connected ^ A . 

with a brass stud on the 
top of the box. 

Three switches, A, 
B, and C, so connect 
the coils with the bind- 
ing screws a and 6 that 
a current can be sent 
through any three coils 
at the same time by 
moving the switches on 
to the proper studs. 

The resistance in ohms of each coil is marked on the box near its stud. 
When the three switches rest upon studs marked 0, the current meets 
with no appreciable resistance in passing through the box, but any 

1 Each additional switch with its corresponding coils increases the range about 
tenfold, so that the range of the instriiment may be very much increased. 

Fig. 390. 



Fig. 391. 

desired resistance within the range of the instrument can be introduced 
by moving the switches on to the studs, the sum of whose resistances is 
the resistance required. This instrument we shall call a resistance box. 

473. Wheatstone bridge. 

Fig. 391 represents a perspective view of the bridge (as modified 
by the author), and Eig. 392 represents a diagram of the essential 
electrical connections. ^ The battery wires are connected with the 
bridge at the binding screws B B'. A galvanometer, G^ is con- 
nected at GG', a resist- 
ance box, r, at K E, and 
the conductor x, whose 
resistance is sought, at 

When the circuit is 
closed by means of the 
key T, the current, we 
will suppose, enters at B ; 
on reaching the point A it divides, one part flowing via the branch 
A G B', and the other via the branch A D B'. If points D and G 
in the two branches be at 
different potentials and a 
connection be made be- 
tween them through the 
galvanometer, (9, by clos- 
ing the key S, there will 
be a current through this 
wire and through the gal- /qq^^- 
vanometer, and a deflec ^—^ 
tion of the needle will be /Q CIh- 
produced. But if the 
points D and G be at 
the same potential, there 
will be no cross current 
through the bridge wire 
and no deflection. Now 
it can be demonstrated 
that points D and G will be at the same potential when R (the re- 

1 The student will find descriptions of the more elaborate bridges and resistance 
coils in such works as Gordon's Electricity and Magnetism, Vol. I, Sylvanus Thomp- 
son's Lessons in Electricity and Magnetism, and in various laboratory manuals. 



sistance) of A D : R of D B' : : R of A G : R (the unknown resistance) 
of G B'. Between A and D and A and G there are three coils of 
wire having resistances respectively of 1, 10, and 100 ohms. One or 
more of these coils are introduced into the circuit by removing the 
corresponding plugs a, &, c, cZ, e, and /. As the other connections 
between A and D, and A and G, have no appreciable resistance, 
being for the most part short brass bars, the only practical resist- 
ance between these points is that introduced at will through the 
coils. Similarly between points D and B^, the only practical resist- 
ance is that introduced at will through the resistance box, and 
between the points G and B' the resistance is the resistance (x) 

It is apparent, then, that in using the bridge after the connec- 
tions are properly made through the several instruments and certain 
known resistances are introduced between A and D, and A and G, 
we have simply to regulate the resistance through the resistance 
box so that there will be no deflection in the galvanometer ; then 
we are sure that the above proportion is true. The first three 
terms of the proportion being known, the fourth term, which is the 
resistance sought, is computable. i 

If the same resistance be introduced between points A and 
G as between A and D, it is evident that the resistance in 
the -resistance box r must be made equal to the unknown 
resistance x in order that there may be no deflection in the 
galvanometer. Consequently when this result is obtained 
the resistance of x may be read from the resistance box. 

Experiment. — Measure the resistance of each of the several spools of 
wire used above, — electro-magnets, electric lamps, etc., — using the 
bridge. Place the switches of the resistance box on the zero studs. 
Make connections as in the description above. Then close the circuit at 

1 The accuracy of the results obtained largely depends upon so choosing resist- 
ances of the bridge as to make the arrangement have maximum sensibility, and upon 
the sensitiveness of the galvanometer. In using the bridge the following directions 
should be observed : (1) Always close the circuit at T before closing the bridge at S, 
and in breaking the circuit reverse this order. (2) Introduce between A and D, and 
A and G, resistance as nearly equal to the resistance sought (a;) as practicable. If 
you have no conception what the unknown resistance is, it is best to begin by using 
high resistances. (3) Use a sensitive galvanometer, e.g. a mirror galvanometer, or 
the galvanometer shown in Fig. 389, substituting the astatic needle for the tangent 



T, and afterwards the bridge at S. There will probably be a deflection 
in the galvanometer. Regulate the resistance through the resistance 
box, throwing in or taking out resistance according as one or the other 
tends to reduce the deflection (the process is much like that of weighing), 
until there is no deflection. Then compute the resistance sought accord- 
ing to the above proportion. 

474. Measurement of galvanometer resistance. Lord Kelvirv's 
method. — The bridge may be used for measuring the resist- 
ance of the galvanometer actually in 
use. The bridge is arranged as in 
Fig. 393. The resistance in the re- 
sistance box K is then varied until 
the deflection of G does not change 
when the key S is closed ; then 

r = K-y 

in which r is the resistance of the 
galvanometer, K is the resistance in 
the resistance box, and a and h are the 
resistances in the arms A G' and A D 
respectively. If a = h, then r = R. 

475. Battery resistance. Mance's Tnethod. 

No definite meaning can be attached to the expression battery 
resistance, since this resistance is complicated by variation in the 
polarization, and this in turn is dependent upon the strength of the 
current, external resistance, etc. Hence the numerous methods 
depending on Ohm's law that have been devised for deducing battery 
resistance are of little value in many cases. 

The method known as Mance's method is one of the best for 
measuring battery resistance, since it requires the battery to be 
constant only during the short time the key is closed. A Wheat- 
stone bridge is arranged as shovm in Fig. 394. The resistance in 
the resistance box E is adjusted until, on pressing down the key 
A, the deflection of the galvanometer does not change ; then 

r = R 


in which r is the battery resistance, R is the resistance in the re- 
sistance box ; a and h are the resistances respectively in the arms 
A G' and AD. If a = 6, the for- 
mula is simplified, and becomes 
r = R. 

The information generally re- 
quired in practice, however, is 
what available difference of po- 
tentials can be obtained with a 
certain working resistance in the 
external circuit. This can be 
obtained by connecting the ter- 
minals of the battery by the 
body offering this resistance, and 
measuring the difference of po- 
tentials between these points by 
means of a potential galvanom- 
eter. If we call this difference 
of potentials V, and the E.M.F. 
of the battery when on open circuit E, and R 
then we may write 

E ^ I ^ p 

R + r R ' 

Fig. 394. 

external resistance, 

where r is such a quantity as satisfies this equation. In other 
words, tills quantity may be taken as the resistance of the battery 
for the current C. 

Section XI. 



476. Electro-motive force of different cells. — If a galva- 
nometer be introduced into a circuit with different battery cells, 
e.g. Bunsen, Daniell, G-renet, etc., very different deflections 
will be obtained, showing that the different cells yield cur- 
rents of different strengths. This may be in some measure 
due to a difference in their internal resistance, but it is chiefly 
due to the difference in their electro-motive forces. We 


learned (§ 431) that difference of electro-motive force is 
due to the difference of the chemical action on the two plates 
used, and this depends npon the nature of the substances 
used. It is wholly independent of the size of the plates ; 
hence the electro-motive force of a large battery cell is no 
greater than that of a small one of the same kind. Conse- 
quently any difference in strength of current yielded by 
battery cells of the same kind, but of different sizes, is due 
wholly to a difference in their internal resistances. 

The electro-motive forces of the Bunsen, Daniell, and 
Grenet cells are respectively about 1.8, 1, and 2 volts. 

477. Divided circuits ; shunts. 

Experiment 1. — Make a divided circuit as in Fig. 395 (using double 
connectors a and 6). Insert a galvanometer, G, in one branch and a 
resistance box, K, in the other. When the current 
reaches a, it divides, a portion traversing one branch 
through the galvanometer, and the remainder passing 
through the other branch and the resistance box. The 
branch a R 6 is called a sliunt or derived circuit. In- 
crease gradualljT^ the resistance in the resistance box. 
The result is that it throws more of the current 
through the galvanometer, as shown by the increase of 
Fm. 395. deflection. 

In a divided circuit the current divides between the paths 
inversely as their resistances. For example, if the resistance 
of the resistance box above be 4 ohms, and the resistance in 
the galvanometer be 1 ohm, then four-fifths of the current will 
traverse the latter and one-fifth the former. 

Suppose that the resistance box and galvanometer be 
removed from the shunts, and that the shunts be of the 
same length, size, and kind of wire, and consequently have 
equal resistances. Using the two wires instead of one to 
connect a and h is equivalent to doubling the size of this 
portion of the conductor ; consequently the resistance of this 
portion is reduced one-half. 



Generally, the joint resistance of two branches of a circuit is 
the product of their respective resistances divided hy their sum. 
478. Kirchhoff's law. 

The following discussion will make the above law evident. First 
it should be understood that when 
a conductor conveys a constant ' j 

current the strength of current 
across all cross - sections of the 
conductor, as A, C, E F, D, and 
B (Fig. 396), is the same. Hence 
the current arriving at C or D of 
the main circuit is equal to the 
sum of the currents which flow 
by the branches ri, r^ and r^- 
This is known as Kirchhoff's First 

By Ohm's law if two points, a 
and &, between which a differ- 
ence of potential Y is maintained, 
be connected by two wires havin^ 

V V 

current in that of resistance r i will be — ' and in the other — ' 

If C be the whole current flowing in the circuit, we have by 
Kirchhoff's law 

V V V 

c=- + - = ^' 

ri To R 

where R is the resistance of a wire which might be substituted for 
the divided conductor between a and 5 without affecting the cur- 
rent. Hence 

\ = — ' and R = — V^ " 

Tx r-2 R ■ ri + Ti 

resistances rx and r2, the 

If there be three separate wires of resistance, as in Fig. 396, we 
shall have in a similar manner 

R Tx Ti Ts 


The reciprocal of the resistance R of a wire, i.e. — ' is called its 
conductivity, sometimes expressed as mhos.'^ We may say, there- 

1 A word formed by writing the word ohin in reverse order. 


fore, in general, when two points in a circuit are connected by a 
multiple arc (a term in common use to denote a divided circuit 
between any two points) consisting of n brandies, the conductivity 
of the multiple arc is equal to the sum of the conductivities of the n 
branches : in other words, the reciprocal of its resistance is equal to 
the sum of the reciprocals of the resistances of its branches. 

479. Shunted galvanometer. — When a current is so strong 
as to produce too violent an impulse upon tlie needle of a 
galvanometer, its terminals may be shunted through a re- 
sistance box, so that any known fraction of the current may 
be deflected through the shunt.. For example, if the shunt 
have a resistance i as great as that of the galvanometer, then 
the current through the latter will be i that through the 
shunt, or yL of the total current. 

480. Methods of comhining cells. 

Experiment 2. — Take two Bunsen cells, and connect the two zinc 
plates by a wire. Then connect each of the carbon plates with a gal- 
vanometer. The E.M.F. of each cell would tend to send a current 
opposite to that of the other cell. But you find that there is either no 
deflection in the galvanometer, or at most a very small one, and this 
shows either that there is no current or that the current is very weak. 
The reason is evident. You have connected two carbons, which have 
the same potential, through the galvanometer ; consequently there should 
be no current between them. The cells are said to be connected in 

A very simple way of showing that a large cell has no 
greater electro-motive force than a small one is to connect 
two such cells in opposition through a galvanometer, or, 
what answers the same purpose, raise the zinc of one of two 
cells of the same size, connected in opposition, nearly out of 
the liquid. The absence of a current shows that the two 
carbons have the same potential, and consequently their 
electro-motive force is the same. 

A number of cells connected in such a manner that the 
currents generated by all have the same direction constitutes 
a voltaic battery. 


The object of combining cells is to get a stronger current 
than one cell will afford. We learn from Ohm's law that 
there are two, and only two, ways of increasing the strength 
of a current. It must be done either by increasing 
the E.M.F. or by decreasing the resistance. So 
we combine cells into batteries, either to secure 
greater E.M.F. or to diminish the internal resist- 
ance. Unfortunately, both purposes cannot be 
accomplished by the same method. 

481. Battei'ies of loiv ■ internal resistance. ~Yig. 
397 represents three cells having all the carbon (+) 
plates electrically connected with one another, and 
all the zinc ( — ) plates connected with one another, 
and the triplet carbons are connected with the 
triplet zincs by the leading-out wires through a 
galvanometer G. 

It is easy to see that through the battery the 
circuit is divided into three parts, and consequent- 
ly the conductivity in this part of the circuit, ac- 
cording to the principle stated in § 478, must be 
increased threefold ; in other words, the internal 
resistance of the three cells is one-third of that of a single 
cell. This is called connecting cells " in multiple arc," and 
the battery is called a " battery of low internal resistance." 
The resistance of the battery is decreased as many times as 
there are cells connected in multiple arc, but the E.M.F. is 
that, of one cell only. 

The formula for the current strength in this case is written 

Fig. 397. 


C = 




in which n represents the number of cells. It is evident 

from this formula that when E is so great that - is a small 



part of the whole resistance of the circuit, little is added to 
the value of C by increasing the number of cells in multiple 

482. Batteries of high internal resista^ice and great E.M.F. 
— Fig. 398 represents four cells having the carbon or + plate 
of one connected with the zinc or — plate of the next, and 

the -|- plate at one end of the 

series connected by leading-out 

wires through a galvanometer 

with the —plate at the other 

m I ■ I 1 I ■ ^^^^ ^^ *^^ series. It is evi- 

1+ - 1+ " I"'' ~ I"*" ^^^^* ^^^^ *^^ current in this 

■ ■ ' ■ series traverses the liquid four 

times, which is equivalent to 
lengthening the liquid conductor four times, and of course 
increasing the internal resistance fourfold. But, while the 
internal resistance is increased, the E.M.F. of the battery is 
increased as unany times as there are cells in series. The gain 
by increasing the E.M.F. more than offsets, in many cases 
(always when the internal resistance is a small part of the 
whole resistance of the circuit), the loss occasioned by in- 
creased resistance. 

The formula for current strength in this case becomes 

It is evident that C is increased most by adding cells in 
series when n r is smallest in comparison with E. 

483. Best arrangement of cells. 

Experiment 3. — Introduce into circuit witli a single cell a resistance 
box and a galvanometer. Throw a resistance of (say) 50 ohms into the 
circuit by means of the resistance box. Note the deflection. Then add 
another cell, in series, to the cell already in use. The deflection is con- 
siderably increased. Other cells -may be added with similar results. 

Experiment 4- — Connect the two cells in multiple arc, keeping the 


same resistance in the resistance box. The deflection is only a very little 
greater than that caused by a single cell. 

Experiment 5. — Connect a single cell with a galvanometer of low 
resistance, so that the whole external resistance may be less than the 
resistance of the single cell. Note the deflection. Then introduce an- 
other cell in multiple arc. The deflection is considerably increased. 

Experiment 6. — Connect the same cells in series. The deflection 
differs but little from that produced by a single cell. 

Hence, (1) wheii the external resistance is large, connect cells 
in series ; (2) luhen the external is less than the internal re- 
sistance, connect cells in multiple arc. 

The two systems may be combined in one battery. Thus 
one pair in series may be placed in multiple arc with another 
pair in series. This combination would give double the 
E.M.F. of a single cell, but the resistance of only one cell. 

With a given number of cells and a given external resistance 
the maxiniuin current is generated when the external and in- 
ternal resistances are er[ual. It is seldom possible in practice 
so to join the cells as to fulfill this condition ; but if the 
strongest possible current be required it should be fulfilled 
as nearly as possible. The fallacy, however, of introducing 
resistance into any part of the circuit for the purpose of 
making these resistances equal must be carefully avoided, 
for resistance wherever introduced can tend only to w^eaken 
the current. Nor must it be supposed that of two batteries 
of equal E.M.F. , but one having a high, the other a low re- 
sistance, the former is better adapted for working through a 
high resistance. It should be borne in mind that the role of 
a battery in general is to maintain a difference of potential 
between its poles, and the element of resistance that it intro- 
duces into the circuit is a necessary evil, not to be voluntarily 

Electro-magnets and galvanometers must he adcipted to the 
circuits in which they are to he placed. In connection with 
the above discussion, it seems proper to introduce, somewhat 


parenthetically, a few facts pertaining to tlie use of electro- 
magnets and galvanometers. 

Li order to produce the greatest effect, the resistance of the 
helix of an electro-magnet should be equal to that of the portion 
of the circuit not included in the helix, i.e. to the rest of the 
circuit. When several electro-magnets are used in the same 
circuit, the sum of the resistances of all the helices should be 
equal to the resistance of the rest of the circuit. 

The same rule applies to galvanometers. High resistance 
galvanometers are most suitable for high resistance circuits, 
and loAv resistance galvanometers are most suitable for low 
resistance circuits. In other words, both galvanometers and 
electro-magnets should be adapted to the resistance of the 
circuit in which they are to be used. 


1. What E.M.F. is required to maintain a current of one ampere 
through a resistance of one ohm ? 

2. Through what resistance will an E.M.F. of ten volts maintain a 
current of 5 amperes ? 

3. What current ought an E.M.F. of 20 volts to maintain through a 
resistance of 5 ohms ? 

4. A voltmeter applied each side of an electric lamp shows a differ- 
ence of potential of 40 volts ; what current flows through the lamp, if it 
have a resistance of 10 ohms ? 

5. The resistance between two points in a circuit is 10 ohms. An 
ammeter shows that there is a current strength in the circuit of 0.5 
ampere ; what is the difference in potential between the points ? 

6. What current will a Bunsen cell furnish when y = 0.9 ohm (about 
the resistance of a quart cell), E = 1.8 volts, and R = 0.01 ohm (about 
the resistance of 3 ft. of No. 16 wire) ? 

[In the following exercises, whenever a Bunsen cell is mentioned it 
may be understood to be a quart cell, having a resistance of about 0.9 
ohm. Its E.M.F. is about 1.8 volts.] 

7. a. AVhen is a large cell considerably better than a small one ? 
h. When does the size of the cell make little difference in the current ? 

8. If you have a dozen quart cells, how can you make them equiva- 
lent to one 3-gallon cell ? 


9. If a battery of 10 cells have an E.M.F. ten times greater than 
that of a single cell, why will not the battery yield a current ten times 
as strong ? 

10. a. The internal resistance of ten cells, connected in multiple arc, 
is what part of that of a single cell ? b. If the cells were connected in 
series, how would the resistance o| the battery compare with that of one 
of its cells? c. How would the E.M.E. of the latter battery compare 
with that of a single cell ? 

11. What current will a single Bunsen cell furnish through an external 
resistance of 10 ohms ? #> 

12. What current will 8 Bunsen cells, in series, furnish through the 
same resistance ? 

SoLOTiox : ^ = io!^^o^/^g = 0.83 + ampere. 

13. What current will 8 Bunsen cells, in multiple arc, furnish through 
the same external resistance ? 

Solution: ^ = ^^^-iL_ = q. 17 + ampere. 

14. What current will a Bunsen cell furnish through an external re- 
sistance of 0.4 ohm ? 

15. What current will a battery of two Bunsen cells, in series, furnish 
through the same resistance as the last ? 

16. What current will two cells, in multiple arc, furnish through the 
same resistance ? 

17. A coil of wire having a resistance of 10 ohms carries a current of 
1.5 amperes. Eequired the difference of potential at its ends. 

18. What would be the resistance at 0° C. of a column of mercury 
154 cm long and f of a square millimeter in cross-section ? 

19. a. What is the resistance of J mile of No. 16 (diam. .05 in.) copper 
wire? b. What E.M.F. will be required to maintain a current of .5 
ampere in this circuit ? 

20. a. The resistance between two points, A and B, of a conductor is 
2.5 ohms ; the resistance of a shunt between the same points is 1.5 ohms ; 
what is the joint resistance between these points ? 6. If a current of 10 
amperes be maintained between these points, what will be the strength 
of current in each branch? c. How would the strength of current 
between these points be affected if the shunt be removed and the same 
fall of potential be preserved ? Why ? 

21. a. Points A and B in a circuit are connected in multiple arc by 
three branches whose respective resistances are 2, 3, and 4 ohms. State 


in order their relative conductivities, b. State the joint conductivity of 
the multiple arc. c. State the joint resistance of the multiple arc. 

22. Four conductors in multiple arc have resistances of 100, 50, 27, 
and 19 ohms. What is their combined resistance ? Ans. 8.3 ohms. 

23. Assume a current of 30 amperes and an E.M.F. of 50 volts ; what 
is the resistance and conductivity ? 

24. A wire is 40 mils (a mil is .001 in.) in diameter, 3 miles long, and 
offers 40 ohms resistance. A second wire of the same material is 50 mils 
in diameter and 9 miles long. What is the resistance of the latter ? 
Ans. 76.8 ohms. 

25. If the terminals of a galvanometer be shunted with a resistance ^^ 
that of the galvanometer, what part of the total current will the galva- 
nometer measure ? 

26. An electric lamp has a resistance of 50 ohms ; it is connected to a 
street main by leads of 2^ ohms resistance. The heat developed in the 
leads is wasted. What portion of the entire heat developed in the derived 
circuit is wasted ? 

27. a. The internal resistance of a voltaic circuit is 2 ohms, and the 
external resistance is 16 ohms ; what portion of the entire heat developed 
in the circuit is generated in the battery ? b. If the external resistance 
of this circuit be reduced to .5 ohm, the heat generated in the battery 
will be what part of the total heat developed in the circuit ? 

28. What is the strength of a current which deposits .02 gram of 
copper per minute ? 

29. Suppose that there are a number of cells joined in series but the 
circuit is completed by short, thick, copper leading wires of practically no 
resistance, would any advanta,ge be gained by adding thereto more cells 
in series ? Exx)lain. 

30. A battery of 20 cells is divided into four groups. Each group 
consists of five cells connected in series, and the four groups are con- 
nected in multiple arc. Compare the E.M.F. and the resistance of this 
battery with that of a single cell. 

31. What E.M.F. of a dynamo generator will be necessary to maintain 
a 12-ampere current for 100 arc lamps in series, each of which has a 
resistance of 5 ohms, the resistance of the line wire being 20 ohms, and 
the dynamo resistance being 25 ohms ? 


Section XII. 


484. Relation of current, resistajice, and pote7itial difference. 
This relation is best understood by the aid of the hydraulic 




lb E 


d I 


e |. 


analogue. B (Fig. 399) represents a tank in which water is main- 
tained at a fixed level by means of a pump (battery), while the 
tank discharges through a pipe (conductor) N 0. At equal inter- 
vals glass tubes, serving as manometers (potential meters), rise 
from the discharge tube. The hight to which the water rises in 
each tube indicates the pressure at that point. The pressure falls 
off uniformly toward 0, as 
indicated by the dotted 
line as. The pressure of 
the column a' a is required 
to force the current against 
the resistance of the pipe 
N O. The force urging the 
water from a' to h' along 
the pipe is the difference 
of pressure at a' and h'. 
This might be called the 
water motive-force between 
a' and h'. If a' e' be four 
times d' e', the resistance 
will be four times as great 
and the water-motive force 
between a' and e' will be 

four times as great as that between d' and e'. The fall of pressure 
is the same for each unit of resistance. These facts in hydraulics 
have their exact parallel in the case of electric flow, and the 
analogues are too apparent to require rehearsal. The fact to be 
established is that every current is due to a determinable E.M.F. 
in the circuit, and fractional parts of the circuit require fractional 
parts of the total E.M.F. The portion of the total E.M.F. exerted 
in forcing the current through any section of the circuit is in exact 
proportion to the relative resistance of such section. For example, 
a battery of ten units resistance may supply an outer circuit of ten 
units resistance; then one-half of the E.M.F. will be exerted 
against battery resistance, and one-half against external resistance. 

Fig. 399. 



If the same battery supply a line of 1000 units resistance, the 
energy expended in the outer circuit is about ^-^% of the total 
energy. The electrical efficiency of an electric generator is the 
resistance of the outer circuit divided by the total resistance. In 
the example above the efficiency of the battery is 

1000 + 10 ' 
485. Expenditure of eneryg. 

or about 

Fig. 400. 

The relative resistances 
of conductors carrying con- 
stant currents define the 
relative expenditure of en- 
ergy upon such conductors. 
The energy may appear as 
heat, as mechanical v^ork, 
or as that of chemical 
decomposition. The vs^ork 
done is due to the current 
and to a fall of potential 
along a conductor, and the 
fall is determined by the 
relative resistances. 

This subject may be il- 
lustrated graphically thus : 
Let the line A B (Fig. 400) 
represent the length of a 
circuit (say 1000 ft.), and 
the line A C the total fall 
of potential ; then obvious- 
ly the slope of the line 
CB will represent the 
average rate of fall of po- 
tential throughout the cir- 
cuit. But suppose that 
the line for equal lengths 
of the conductor varies in 
resistance. Thus assume 
that one-tenth the resist- 
ance and consequently one- 
tenth the fall of potential 
(Cd) is included in the 



first quarter (A a), or 250 feet; then that the next 250 feet (a 6, 
being very fine wire, perchance) represents one-half the total resist- 
ance; the next 250 feet (be) represents one-fourth the total resist- 
ance ; and the remaining resistance, fifteen-hundredths, is in the 
last section (c B) of 250 feet. Then the lines C a', a' 6', h' c\ and 
& B represent respectively the rat.e of fall of potential in each sec- 
tion of 250 feet. The fall in each section is proportional to the 
resistance, and again to this is proportional the v^ork performed by 
the current in each section. 

The above facts may be verified experimentally as follows : P Q 
(Fig. 401) is a fine German-silver wire one meter long, stretched 
along a board over a metric scale ; B is a battery ; G, a mirror 
galvanometer ; K, a key for closing the circuit ; and R a coil of 
German-silver wire of high resistance. When the free end of the 
wire, S., is applied to any 
point of the wire PQ, the 
circuit is shunted through 
G and R. By this means 
an extremely small por- 
tion of the current will be 
shunted through the gal- 
vanometer. This will not 
perceptibly change the po- 
tentials of the points P and 
S. Observations are made 
by placing the free end of 
the wire, S, at different 

points along the wire and reading the deflection produced at each 
point. Kow if the potential differences are proportional to the 
resistances, i.e. to the lengths of wire between the points P and S, 
it follows that this resistance should have a constant ratio to the 
difference of potential between the same points. But the difference 
of potential is expressed by the current which it sends through the 
galvanometer, so that this current (which is indicated by the deflec- 
tion of the galvanometer needle) should be proportional to the 
distance between P and S. 

E.M.F. of a battery. The E.M.F. of a battery is considered to 
be the difference of potential between its poles when the circuit is 
open. If the circuit be closed, the difference of potential at its 
poles will depend on the resistance of the conductor connecting 
them. In this case it is 

Fig. 401. 


in which E is the E.M.F. of the battery, R the external resistance, 
and r the battery resistance. From this it follows that (1) the 
difference of potential at the poles of the battery is less when the 
circuit is closed than when it is open ; (2) it is less with a small 
than with a large external resistance ; and (3) it is greater with a 
small than with a large internal resistance. 

Section XIII. 


486. Law of magnets} — Suspend by fine threads in a 
horizontal position two stout darning-needles which have 
been drawn in the same direction {e.g. from eye to point) 
several times over the same pole of a powerful electro-magnet. 
These needles, separated a few . feet from each other, take 
positions parallel with each other, and both lie in a northerly 
and southerly direction with the points of each turned in the 
same direction. 

That point in the Arctic zone of the earth towards which 
magnetic needles point is called the north magnetic pole of 
the earth. That end of a needle which points toward the 
north magnetic pole of the earth is called the north-seeking, 
marked, or -\- jpole ; this is the end that is always marked for 
the purpose of distinguishing one from the other. That end 
of the needle which points southward is called the south- 
seeking, unmnrked, or — jmle. 

Experiment 1. — Bring both points near each other ; there is a mutual 
repulsion. Bring both eyes near each other ; there is a mutual repulsion. 
Bring a point and an eye near each other ; there is a mutual attraction. 

1 The word magnet is supposed to have been derived from the name of an ancient 
district in Asia Minor called Magnesia, where was discovered at an early period a 
mineral (noAV known as the magnetic oxide of iron) which possesses the property of 
attracting iron. The term lodestone, or " leading stone " (inasmuch as it takes a 
definite position north and south) has been given to these natural magnets. 


Like poles of magnets repel, unlike poles attract each other. 
487. Magnetic transparencij and induction. 

Experiment 2. — Literpose a piece of glass, paper, or wood-shaving 
between the two magnets. These substances are not themselves per- 
ceptibly affected by the magnets, nor do they in the least affect the 
attraction or repulsion between the two magnets. 

Substances that are not susceptible to magnetism are said 
to be magnetically transparent. When a magnet causes another 
body, in contact with it or in its neighborhood, to become a 
magnet, it is said to induce^ magnetism in that body. As 
attraction, and never repulsion, occurs between a magnet and 
an unmagnetized piece of iron or steel, it must be that the 
magnetism induced in the latter is such that opposite poles 
are adjacent ; that is, a N" or +pole induces a S or — pole 
next itself, as shown in Fig, 402. 

Fig. 402. 

488. Polaritij. 

Experiment 3. — Strew iron filings on a flat surface, and lay a bar 
magnet on them. On raising the magnet it is found that large tufts of 
filings cling to the poles, as in Fig. 403, especially to the edges ; 
but the tufts diminish regularly in size from each pole towards 
the center, where none are found. 

Magnetic attraction is greatest at the poles, and 
diminishes toiaards the center, lahere it is nothing ; i.e. 
the center of the bar is neutral. This dual character 
of the magnet, as exhibited at the opposite extremi- 
ties of a magnet, is called polarity. If a magnet be 
broken, each piece becomes a magnet with two poles 
and a neutral line of its own. 

489. Retentivity and resistance. 
It is more difficult to magnetize steel than iron ; 

on the other hand, it is difficult to demagnetize steel, 

1 A word first vaguely used by Faraday in ttie sense of influence. 

Fig. 403. 



while soft iron loses nearly all its magnetism as soon as it is 
removed from the influence of the inducing body. That 
quality of steel by which it resists the escape of magnetism 
which it has once acquired is called its retentivity. The 
greater the retentivity of a magnetizable body, the greater is 
the resistance which it offers to becoming magnetized. The 
harder steel is, the greater is its retentivity. Hence, highly 
tempered steel is used for permanent magnets. Hardened 
iron possesses considerable retentivity ; hence the cores of 
electro-magnets should be made of the softest iron, that they 
may acquire and part with magnetism instantaneously. 

490. Forms of artificial magnets. — -Artificial magnets, in- 
cluding permanent magnets and electro-magnets, are usually 
made in the shape either of a straight bar or of the letter U, 
according to the use to be made of them. If we wish, as in 
the experiments already described, to use but a single pole, 
it is desirable to have the other as far away as possible; 
then, obviously, the bar magnet is most convenient. But if 
the magnet is to be used for lifting or holding weights, the 
U-form is far better, because the attraction of both poles 
is conveniently available. Magnets, when not in use, ought 
always to be protected by armatures (A, Eig. 404) of soft 
iron ; for, notwithstanding the retentivity of 
steel, they slowly part with their magnetism. 
But when an armature is used, the opposite poles 
of the magnet and armature being in contact 
with each other, 'i.e. '^ with S, they serve to bind 
each other's magnetism. Thin bars of steel can 
be more thoroughly magnetized than thick ones. 
Hence, if several thin bars (Fig. 404) be laid side 
by side, with their corresponding poles turned in 
the same direction and then screwed together, a 
very powerful magnet is the result. This is called a com- 
fo^ind magnet. 

Fig. 404. 



Section XIV. 


These lines are easily 
The field of force around 

491. Magnetic lines of force. 
studied by the use of iron filings 
a magnet is shown 
by placing a paper 
over it, dusting 
filings upon the 
paper, and tapping 
it. The filings 
take symmetrical 
positions, form 
curves between the 
poles of the mag- 
net or magnets, 
and show that the 
lines of force con- 
nect the o][)j)osite 
poles of the mag- 
net. The fact is, that each filing, when brought within the 
influence of the magnetic field, becomes a magnet by induc- 
tion, and of necessity tends to take a definite position which 

Fig. 405. 


'^ / 

— > 


:: : 

S )?^>- 

/ilWV^^- -■ 


Fig. 406. 



represents the resultant of the forces acting upon it from 
each pole of the system. Eig. 405 represents a magnetic 
field photographed from a specimen paper, and Eig. 406 is a 

,/ / 

^ \ 


Fig. 407. 

graphical representation of the same. In this illustration 
the unlike poles of two magnets are placed opposite each 
other. Fig. 407 is a diagram of paths of lines of force of a 
bar magnet, and Fig. 408 of a horseshoe magnet. 

492. Magyietic circuit. — A line of 
force is assumed arbitrarily to start 
from the N-pole and to pass through 
the surrounding medium (e.g. the air), 
entering the magnet by the S-pole, 
and completing its path through the 
magnet itself to its starting-point 
(the IST-pole), thus forming a complete 
circuit (Fig. 407). These lines do 
not all emerge, however, from the 
extremities. A multitude of lines 
start from all parts of the magnet 
and enter at corresponding points on 
the other side of its central or neutral 
line. No magnetic line of force can exist without completing 

y / I 

Fig. 408. 


its own circuit/ and lines of force never cross or merge into 
one another, consequently a magnet cannot have a single pole. 
Lines of force possess several peculiar characteristics. One 
is that in air and most other mediums they tend to separate 
from one another, but at the same time tend to become as 
short as possible. The strain is as if these lines were 
stretched elastic threads endowed with the property of repel- 
ling one another as well as of shortening themselves ; in 
other words, there is tension along the lines and pressure at 
right angles to them. Air is a poor conductor for lines of 
force, or its 'permieability'^ is low ; on the other hand, iron 
has high permeability for lines of force, and if a 
piece of iron be brought within a magnetic field, a [ 

portion of the lines of force will crowd together 
into it, leaving their normal paths through the air 
for a medium of greater permeability. 

493. Line of force the axis of ether luhirls, 

A line of force is supposed to represent the axis of 
a series of ether whirls. Fig. 409 gives a crude pic- 
torial representation of the supposed constitution of 
the whirls of an electro-magnetic line of force. A 
series of curtain rings might be strung 
upon a stretched thread and caused to 
rotate around it. This would give some 
idea of the hypothesis ; the thread 
would give the direction of the line of 
force and its conventional representa- 
tion as a simple line. The perfectly cir- 
cular line of force is such as those (§ 445) i 
surrounding a wire carrying a current, 1 
and represented in Fig. 410. Its mechanical analogue ^ 
is seen in a smoke-rino^ such as is sometimes caused to Fig. 409. 

Fig. 410. 

1 Herein exists a notable difference between these and electrostatic lines of force. 
Every electrostatic charge is bound, i.e. has an opposite and equal charge somewhere. 
To this its lines of force go ; but there is no circuit, there is only a connection. 

2 Permeability is a quality christened by Lord Kelvin. 



rise from the bowl of a tobacco pipe by skillful operation. It is 
well illustrated by the rings arising from the spontaneous combus- 
tion of phosphureted hydrogen in air. These show the whirling, 
rotary motion around the circular axis of the ring. Such are called 
vortex rings. 

494. Attraction and rejndsion between magnetic poles. 

The tendency of these whirls is to bulge out by reason of centri- 
fugal force. An assemblage of such parallel whirls would compress 
each other laterally and cause a longitudinal tension (§ 425). On 
this hypothesis the phenomena of magnetic attraction and repulsion 
are explainable. 

If the N-pole of one magnet be placed opposite the S-pole 
of another (Fig. 406), the lines of force issuing from the 
former will enter the latter, and, tending to shorten, will 
produce attraction. If the similar ends be opposed (Fig. 411), 

Fig. 411. 

the lines of force will be turned away from each pole in all 
directions, and will complete their circuits independently. 
Thus becoming parallel they will repel one another; for 
this reason like magnetic poles repel each other. 
495. Equipotential surfaces. 

The potential at any point in a magnetic field is the quantity of 
work that would have to be done in bringing a + unit of magnetism 
from an infinite distance to that point. 

If a magnetized particle be moved either toward or from a mag- 
netic pole, work is done either by or against the attracting or re- 
pelling force, and the particle is said to be moved from a point 
where the magnetic potential has one value to a point where it has 



another value. Imagine an N-pole isolated from its companion 
S-pole. We may suppose that all points at equal distances from 
this pole are at the same potential, and these points joined form a 
spherical surface. The potential at every point of this surface 
(being the same) may be represented by V. Within and without 
this lie successive spherical equipotential surfaces over each of which 
the potential is constant, but the potential at each surface differs 
from that at any of the others. No work is required to move a 
quantity of magnetism from one point on an equipotential surface 
to another point on the same surface. Equipotential surfaces are 
not necessarily spherical. But whatever be their form, the lines of 
force are always at right angles to the equipotential surfaces. 

496. Tubes of force. 

Suppose A B (Fig. 412) to be a section of an equipotential surface. 
Lines of force pass through the surface, those grazing its edges cut- 
ting off an area. A' B', from another equipotential surface. The 
space comprised between these equi- 
potential areas and bounded laterally 
by lines of force is called a tuhe of 
force. This space may be supposed 
to be filled with a bundle of lines of 
force. Now the intensity of the 
magnetic force at the potential A' B', 
as compared with that at A B, is in 
inverse proportion to the magnitudes 
of their respective areas. 

The intensity of magnetic forces 
at any two points may be compared 
by stating the relative numbers of 
the lines of force which pass through 
equal units of area of the equipoten- 
tial surfaces containing the points 

compared. The fewer these lines per unit area, the less the local 
intensity of the force. The strength of field between the poles of a 
magnet may be expressed in dynes, but now it is more common 
among electricians to express this strength by the number of lines 
of force per cm^, each line of force representing a dyne. 

For example, if the strength of the magnetic field, or the force 
on a unit pole, be ten dynes at any point, then ten lines of force are 

Fig. 412 



said to pass through an area of 1 cm^ held perpendicularly to these 

497. Strength of magnetic field by the method of oscAllatioiis. 

This method of comparing the strength of the magnetic field at 
different points in it depends on the principle that a pendulum 
makes isochronous oscillations when the length of the arc is very 
small, and that the force which is always pulling it back to its 
position of rest is proportional to the square of the number of oscil- 
lations in a given time. 

A magnetic needle about 3 in. long is suspended by a silk fiber 
in the magnetic meridian. The needle is deflected from the merid- 
ian and allowed to vibrate, and the number of vibrations made in 
a given time are counted. Suppose that 11 oscillations be made 
in 30 seconds under the action of the earth. If E represent the 
strength of the earth's magnetic field, then E may be measured by 
112= 121. 

Then on placing a long bar magnet (Fig. 413) in a vertical position 
in the same meridian with the needle, and its N-pole opposite the 
S-pole of the needle, the two poles being 4 cm apart, 
suppose that the number of oscillations made in 30 
seconds be 61. If, then, M4 represent the intensity 
[ of the field at a distance of 4 cm, E + M4 is measured 
by 512 = 2601. Hence M4 is measured by 2601 — 121 
= 2480. 

Suppose that on removing the pole to a distance 
of 8 cm the number of oscillations is 27. Denoting 
by Ms the intensity of the field of the magnet at a 
distance of 8 cm, we have E + Ms measured by 272 = 
729, and therefore Ms is measured by 729 — 121 = 
608. Hence M4 : Ms = 2480 : 608 = 4 : 1 (within the 
Fig. 413. limit of errors of observation). 

498. Latu of inverse squares. — In this manner may be 
demonstrated experimentally the Law of Inverse Squares as 
applied to magnetism, viz. : The force exerted between two 
magnetic poles varies inversely as the square of the distance 
between them. 

The magnitude of the force in any case is numerically equal 


to the product of the strength of the poles divided by the square 
of the distcuice between them; 



The C.G.S. unit pole is that pole ivhich repels an equal pole 
placed a centimeter away ivith a force of one dyne} 

499. Definitions. — We are now in a position to understand 
the following definitions : A portion of space throughout which 
magnetic effects are exerted by a distribution of magnetism is 
called a magnetic field. At each point of the field a pole of 
intensity, m, is acted on by a definite force. . The intensity 
of a magnetic field at a given point is equal to the force in 
dynes with which a unit pole would be acted upon at that 
point. Let H denote the intensity of field at any point ; 
then the force actually exerted at that point on a pole of 
strength m is m H. 

A line of magnetic force is a line drawn in such a manner 
that the tangent to it at each point is in the direction of the 
resultant magnetic force at that point. 

The magnetic potential at any point is the work that must 
be done against the magnetic forces in bringing up a unit 
magnetic pole from a point at an infinite distance to the 
given point. The difference of magnetic ptotential between 
two points in the field is the work done in transferring a unit 
magnetic pole from one of these points to the other, against 
magnetic forces. A surface in which the magnetic potential 
at all its points is the same is an equipotential surface. 

Let m be the quantity of magnetism at one of the poles of 
a magnet, and I the distance between the poles ^ ; the magnetic 
moment of the magnet is the product m I. 

1 In practice it is impossible to obtain a single isolated pole ; the total quantity of 
(+ and. — ) magnetism in any magnet is, algebraically reckoned, zero. 

2 In a steel bar magnet the poles are not strictly at tbe extremities, and hence 
tbe magnetic length is a little less tban tbe actual length of the magnet. For most 
bar magnets, the magnetic length is about .83 of the actual length. The magnetic 
length of a horse-shoe magnet is the shortest distance between its poles. 



The intensity of mag7ietization is the ratio of the magnetic 
moment of a magnet to its volume. 

A magnetizable body placed in a magnetic field becomes 
magnetized in the direction of the lines of force of the field. 
Its magnetism is called induced magnetism, and the action 
itself is called rnagnetic induction. The magnetism retained 
by a magnetic body after it has been withdrawn from the 
field is called residual magnetism. 

Section XV. 


500. The earth a magnet. 

A dipping-needle is so supported that it can revolve in a vertical 
plane. Indifferent equilibrium is first established in the steel 
needle, so that if placed in a horizontal (or any other) position it 
will rest in that position. Then it is strongly magnetized. After- 
ward it will take the horizontal position only at the magnetic 
equator of the earth. 

Experiment 1. — Place a dipping-needle over the + pole of a bar 
magnet (Fig. 414). The needle takes a vertical position with its — pole 
down. Slide the supporting stand along the bar ; the — pole gradually 

rises until the stand reaches 
Nra K, - c ^s the middle of the bar, 

where the needle becomes 
horizontal. Continue mov- 
ing the stand toward the 
S — pole of the bar ; after 
passing the middle of the 
bar the + pole begins to 
dip, and the dip increases until the needle reaches the end of the bar, 
when the needle is again vertical with its + |)ole down. 

If the same needle be carried northward or southward along the 
earth's surface, it will dip in the same way as it approaches the polar 
regions, and be horizontal only at or near the equator. 

Fig. 414. 



Experiment 2. — Support a small pane of window glass on a table, by 
placing under the glass near its corners four slices of cork about one- 
eighth of an inch thick. Be- 
neath the center of the glass 
on the table place a chcular 
disk of magnetized steel. Sift 
iron turnings upon the upper 
face of the glass through a 
fine wire sieve. Gently tap 
the glass at convenient points 
with the end of a lead-pencil. 
The filuigs arrange themselves 
in Imes radiating from each 

Experiment 3. — Suspend 
a small magnetized cambric 
needle by a fine thread at its 
center and carry it around 
the disk (Fig. 415). The 
needle passes through all the 
phases stated in Exp. 1, so 
that we may fancy the disk 

to be the earth, and study therefrom, in a general way, the changes 
that the needle undergoes, as it is carried around the earth in a norther- 
ly or southerly direction. 

Fig. 415. 

The last experiment presents a true exhibition^ on a small 
scale, of what the earth does on a large one, and thereby 
presents one of many phenomena which- lead to the conclu- 
sion that the earth is a magnet. In other words, these 
phenomena are just what we should expect if a huge magnet 
were thrust through the axis of rotation of the earth, as 
represented in Fig. 416, — having its jST-pole near the S geo- 
graphical pole, and its S-pole near the N geographical pole \ 
or if the earth itself were a magnet. 

501. Magnetic poles of the earth. — It will be seen that 
there are two points where the needle points directly to the 
center of the disk. A point was found on the western coast 



of Boothia, by Sir James Ross, in the year 1831, where the 
dipping-needle lacked only one-sixtieth of a degree of pointing 

directly to the 
earth's center. The 
same voyager sub- 
sequently reached 
a point in Victoria 
Land where the 
opposite pole of 
the needle lacked 
only 1° 20' of 
pointing to the 
earth's center. 

It will be seen 
that, if we call that 
end of a magnetic 
needle which 
points north the 
N-pole, we must 
call that magnetic pole of the earth which is in the northern 
hemisphere the S-pole, and vice versa. (See Fig. 416.) Hence, 
to avoid confusion, many careful writers abstain from the use 
of the terms north and south poles, and substitute for them the 
tQYm^ positive and negative, or marked and unmarked poles. 

502. Variation of the needle. — Inasmuch as the magnetic 
poles of the earth do not coincide with the geographical poles, 
it follows that the needle does not in most places point due 
north and south. The angle which the vertical plane through 
the axis of a freely suspended needle makes with the 
geographical meridian of the place is known as the angle of 
declination. In other words the angle of declination is the 
angle formed by the magnetic and geographical meridians. 
This angle differs at different places. The magnetic axis of 
a needle is a straight line connecting its two poles. 

Fig. 416. 



Experiment 4. — We can easily find, as did Columbus, the declination 
at any place by the following method : Set up two sticks so that a string 
joining them will lie in the same vertical plane with the Pole Star ; the 
string will lie in the geographical meridian. Place a long magnetic 
needle over the string ; the angle between the needle and the string is the 
required declination. K great accuracy be required, allowance must be 
made for the fact that the star is not exactly over the pole, but appears 
to describe daily around it a circle whose diameter is at the present time 
about 2i°. 

503. Isotonic curves. — These are lines connecting all points 

Fig. 417. 

of equal declination on the earth's surface. The line of no 
declination or isogonic of 0° (Fig. 417) commences at the N. 
magnetic pole about lat. 70°, long. 96°, passes in a south- 
easterly direction across Lake Erie and Western Pennsyl- 
vania, and enters the Atlantic Ocean near the boundary 
between the Carolinas. Pursuing its course through the 
south polar regions, it reappears in the eastern hemisphere 
and crosses Western Australia, the Caspian Sea, and thence 
to the Arctic Ocean. There is also a detached line of no 
declination inclosing an oval area in Eastern Asia and the 
Pacific Ocean. In the eastern (or Atlantic) hemisphere, 



bounded by the line of no decimation, the declination is 
westward, as indicated by continuous • lines in the figure. 
In the western (or Pacific) hemisphere the declination is 
eastward, as indicated by dotted lines. 

The magnetic poles are not fixed objects that can be located 
like an island or cape, but are constantly changing. They 
appear to swing, something like a pendulum, in an easterly 
and westerly direction, each swing requiring centuries to 
complete it. The north magnetic pole is now on its westerly 

504. Inclination or dip. — We have seen that in the northern 
hemisphere the lines of force tend downward and northward. 

A magnetic needle thus 
tends to place itself so 
that its axis points down- 
ward and to the magnetic 
pole of the earth. The 
angle which the axis of 
a freely suspended mag- 
netic needle makes with 
the horizontal plane is 
called the inclination or 
dip of the needle. Fig. 
418 represents a dipping- 
needle such as is used 
in determining magnetic 
inclination, and Fig. 419 
represents a declinometer 
for determining the decli- 
nation. A is a mounted 
telescope for sighting 
north star. Its axis is 

Fig. 418. 

some astronomical object, e.g. the 
levelled by the spirit level B. 

The line passing in an easterly and westerly direction 



Fig. 419. 

around the earth along which the lines of force (or needle) 
are horizontal, i.e. at which 
the dip is zero, is the magnetic 
equator. It does not coincide 
with the geographical equator. 
The lines roughly parallel to 
the magnetic equator, along 
Avhich the dip is equal, are 
the magnetic parallels. These 
are lines along which equipo- 
tential surfaces cut the sur- 
face of the earth. 

We have before noted the 
fact that lines of force are 
always at right angles to equi- 
potential surfaces, and conse- 
quently at the magnetic poles 
where the dip is 90° the equipotential surfaces are tangent 
to the earth's surface. 

505. Intensity. — The force, expressed in absolute measure, 
with which the earth's magnetism urges a unit magnetic pole 
(§ 498) is the intensity of the earth's magnetic field ^ at the 

The earth's action on a needle is a mechanical couple, 
the effect of which is to cause only a rotary motion. This 
is what is meant when the earth's action on the needle is 
said to be directive only. 

506. Connection betiveen the sun and the earth^s magnetism. 

1 If the inclination be found, and the horizontal component of the intensity of 
the earth's field acting upon a unit pole be known, we have the data reqiiired for 
determining by parallelogram of forces the whole intensity of the earth's magnetic 
field in the direction of the lines of force at any point. Such a measurement is of 
importance in refined work, since it consists essentially in determining the couple 
which must be exerted by the earth's magnetic force on a needle in order to balance 
that produced by the current. For methods of determining these quantities, see 
Cumming's Electricity, or any complete laboratory manual. 



Magnetic storms, or disturbances of the earth's magnetism 
coincident with outbreaks of sun spots and solar storms, point 
to an undoubted connection or sympathy between the sun and 
the earth's magnetism ; but of the nature of this connection 
our knowledge is as yet very limited. 

Section XVI. 


507. Magnetic field due to a circular current. — If a wire 
be bent into the form of a circle of about 10 in. diameter, 
and placed vertically in the magnetic meridian, and a card- 
board be placed at right angles to the circle so that its hori- 
zontal diameter is coincident with the upper surface of the 
cardboard, and a very strong current be sent through the 
wire in the direction indicated by the arrow-head in the wire, 
iron filings sifted upon the card will arrange themselves as 
shown in Fig. 420. And if a freely suspended test-needle 

Fig. 420. 

be carried inside and outside the circle, the several positions 
taken by the needle, as indicated in the figure by arrows, 
corroborate the directions of the lines of force as indicated 
by the filings. 



If the direction of the current be reversed, the direction of 
the needle will be reversed wherever it may be placed. 

The direction indicated by the is'-pole of the magnetic 
needle placed anywhere in the field is called the positive 
direction of the lines of force. 

508. Magnetic current. — In fact, when a current traverses 
a wire (or other conductor) lines of force encircle the electric 
current at right angles to it.^ The electric current and its 
encircling lines of force always co-exist, and one varies directly 
as the other when there is no magnetic substance near the 
wire. The direction of the encircling lines of force with 
reference to the electric current may be illustrated by the 
use of a corkscrew. The direction of the electric current 
corresponds to the propul- 
sion of the point of the 
corkscrew when entering 
a cork, and the direction 
in which the screw is 
turned or the hand is 
twisted in propelling it 
corresponds to the direc- 
tion of the lines of force. 

If a circle of wire bear- 
ing a very strong current 
be freely suspended and a ^^^- ^'^'^• 

pole of a very strong bar magnet be presented to one of its 

1 " Every conducting wire is surrounded by a sort of magnetic whirl. A great part 
of the energy of the so-called electric current in the wire consists in these external 
.magnetic whirls. To set them up requires an expenditure of energy ; and to main- 
tain them requires a constant expenditure of energy. It is these magnetic whirls 
which act on magnets, and cause them to set, as galvanometer needles do, at right 
angles to the conducting wire." — S. P. Thompson. 

It may be demonstrated that a law analogous to Ohm's Law Is applicable to the 
magnetic whirl or "flux." Let C=:the strength of the magnetic flux (or field) 
expressed in lines of magnetic force ; M = the force which gives rise to the magnetic 
flux, called the magneto^notive force; and 11=: the resistance to the magnetic flux ; 




faces, the circle will be attracted or repelled according to "the 
nature of the pole and the direction of the current. We may 
consider a circular current as if it were a very short magnet, 
one face of the circle being the north end, and the other face 
the south end, as represented in Fig. 421. 

Observe (Fig. 420) that the directions of the filings near 
the center of the circle lie nearly parallel with its axis, but 
outside to the right and left of the axis the filings lie in 
curves around each wire. 

509. Solenoid. — An insulated wire wound into a helix of 
considerable length as compared with its diameter is called 

a solenoid} It is evident 
that the intensity of the 
magnetic field must be 
greatly increased by the 
joint action of the many 
current turns. The mag- 
netic field within the sole- 
noid is nearly uniform in 
strength, and the lines of force to within a short distance of 
its ends are parallel with its axis, as shown in Fig. 422.^ 

510. Magnetic polarity of electro-magnetic solenoid. — Fig. 
423 represents a small battery floating on water. The lead- 
ing wire of the cell is 
wound into a horizon- (#oouoouo^ ,^ 
tal solenoid. Slowly 
after the cell is floated 
it will take a position 
so that the axis of 
the solenoid will point 
north and south like fj^. 423. 

Fig. 422. 

1 Faraday first applied the term solenoid to a system of circular currents parallel 
with one another. 

2 An open solenoid of a single layer is here given, in order the better to show the 
directions of the several lines of force. 


2 I llllllMHIliP 

Fig. 424. 

a magnetic ceedle. Hold (say) the S-pole of a bar magnet 
near that end of the solenoid which points north; the solenoid 
is attracted by the magnet. Hold the N-pole of the magnet 
near the north-pointing end of the solenoid; the magnet 
repels the solenoid. 

Eepeat the above, using in place of the bar magnet another 
solenoid (Fig. 424) ; there will 
be a repetition of the same phe- 
nomena as obtained with the bar 
magnet. Introduce a rod of soft 
iron into the solenoid held in the 
hand, thereby making of it an 
electro-magnet; the only change 
observed ■ is that the force of 
attraction and repulsion is greatly increased. 

Place the wire of another battery over and parallel with 
the coil (Fig. 425), so that the two currents will flow in planes 

at right angles with 
^y >^? — ^ — ^^'^ each other. The 

coil is deflected like 
a magnetic needle 
(Fig. 426). 

Eeverse the di- 
rection of the cur- 
rent above and the 
deflection is reversed. 

We thus prove that a solenoid bearing a current possesses 
polarity as if it were a magnet, and that there can be pro- 
duced by a current-bearing solenoid a magnetic field of the 
same character as that produced by a permanent magnet. 
There is no essential difference between a permanent magnet, 
a current-bearing solenoid, and an electro-magnet, except that 
the last may be made much stronger than either of the 

Fig. 425. 

Fig. 426. 



Given the direction of the current in a solenoid, to find the 

N~ and S-poles of the solenoid, and vice versa. 

Rule 1. Place the palm of the right hand against the side 

of the solenoid so that the fingers will point in the direction 

of the current passing 
through the windings 
(as shown in Eig. 427) ; 
the thumb will point in 
the direction of the N- 
pole of the solenoid or 

EuLE 2. Ascertain 
the N-pole of the sole- 
noid or electro-magnet 
with a Tnagnetic needle, 

and place the palm of the right hand upon the solenoid so that 

the outstretched thumb points in the direction of the N-pole ; 

the fingers will point in the di7Xction in which the current 

passes in the windings. 


Fig. 427. 

Section XVII. 


511. MutiLal action of currents on one another. — We have 
hitherto discussed the direction of the magnetic fiekl due to 
a straight current, have determined the direction which a 
test-needle takes in virtue of the action of the current field, 
and have learned to regard an electric current as producing 
north and south polarity along the whole circuit of the cur- 
rent-bearing wire. That is, if we suppose that a test-needle 
be moved up or down just back of the current-bearing wires 

1 The following suggestion will often prove of practical value : that is the south 
pole of a helix where the current corresponds to the motion of the hands of a watch, 
S^and that is the north pole where the current is in the reverse direction, *N. 



(Fig. 428), the N- and S-poles will take the positions indi- 
cated by n and s. We may readily premise from inspection 
of the polarity developed, that if the wires were so sus- 
pended as to be free to move either toward or from each 
other, the pair of wires in which the currents flow parallel to 
each other and in the same direction. A, would attract each 
other, and the pair of wires in which the currents flow in 
opposite directions, B, would repel each other ; but if the 
currents be inclined to each other as in Fig. 429, they will 

t t 

t i 

A B 

Pig. 428. 

Fig. 429. 

tend to move into a position in which they will be parallel 
and in the same direction. That such actually takes place 
may be shown by the following experiments : — 

Experiment 1. — Eig. 430 represents a portion of a divided circuit. 
The lower ends of the wires dip about one-sixteenth of an inch into 
mercury, and the wires are so suspended that they are free to move to- 
ward or from each other. Send a current of a battery of three or four 
Bunsen cells, in multiple arc, through this divided circuit. The two 
portions of the current travel in the same direction and parallel with each 
other, and the two Avires at the lower extremities move toward each 
other, showing an attraction. 

Experiment 2. — Make the connections (Fig. 431) so that the current 
will go down one wire and up the other. They repel each other. 



Experiment 3. — Send a current through the spiral wire represented 
in Fig. 432. Here the current flows nearly parallel with itself, and the 
attraction causes the coil to contract and to be lifted out of the cup of 
mercury below. But the instant it leaves the mercury the circuit is 
broken, the current and attraction cease, and the wire dips into the 
mercury again. Thus rapid vibratory motion of the coil is produced. 

In the experiment with the floating cell and current-bearing 
wire placed over and parallel to the solenoid (Fig. 425), a 
careful examination will disclose the fact that not only do 
the planes in which the current flows in the coil tend to 

ff ii t ^j^^ ' "^^ 

Fig. 430. 

Fig. 431. 

Fig. 432. 

become parallel to the current above, but that the current in 
the upper half of the coil, where the influence due to prox- 
imity is greatest, tends to place itself so as to flow in the 
same direction as that of the current above. 

512. Ampere's Laws. — Law 1. Parallel currents, if in the 
same direction, attract one another ; and if in opposite direc- 
tions, they repel one another. 

Law 2. Currents that are not parallel tend to become parallel 
and floiD in the same direction. 

A little reflection will show that the observed motion is 
the expression of a tendency on the part of any movable 
current to cut lines of magnetic force at right angles, the direc- 

ampere's theory of- magnetism. 543 

tion of motion being reversed when the direction either of 
the lines of force or of the current is reversed. This prin- 
ciple is of immense importance from an industrial aspect. 
The most important outcome of its application is the dynamo- 
motor (§ 533) by means of which electrical energy is converted 
into TYieclianical energy, and through the agency of which 
electric street cars are j)ropelled. 

513. Am,jpere^s theory of magnetism. — This celebrated theory 
briefly stated is that magnets and solenoid systems are funda- 
mentally the same; that magnetism is simply electricity in 
rotation, and that a magnetic field is a sort of whirlpool of 
electricity. Not, of course, that a steel magnet contains an 
electric current circulating round and round it as does an 
electro-magnet, but that every molecule of iron, steel, or 
other magnetizable substance is the seat of a separate current 
circulating round it continuously and without resistance, and 
thus every molecule is a magnet. 

According to the theory, in an unmagnetized bar these 
currents lie in all possible planes, and, having no unity of 
direction, they neutralize one another, and so their effect as 
a system is zero. But if a current of electricity or a magnet 
be brought near, the effect of the induction is to turn the 
currents into parallel planes, and in the same direction, in 
conformity to Ampere's Second Law. If the retentivity be 
strong enough, this parallelism will be maintained after the 
removal of the inducing cause, and a permanent magnet is 
the result. 

Intensity of magnetization depends on the degree of paral- 
lelism, and the latter depends on the strength of the influ- 
encing magnet. When these currents have become quite 
parallel, the body has received all the magnetism that it is 
capable of receiving, and is said to be saturated. Although 
the currents really circulate around the individual molecules, 
yet tlie resultant of these forces is essentially the same as if 



a superficial sheet of currents circulated around the body as a 

whole. Fig. 433 represents sections of a cylindrical magnet, and 

the included circles represent 
the circulation of the several 
currents around the mole- 
cules lying in these sections. 
It will be seen that the cur- 
rents at the contiguous sides 
of any two of these circles 
move in opposite directions, 
and therefore must neutral- 
^ ize each other; while the 

^^^' *^^- currents that pass next the 

circumference of the magnet are not so affected. 

514. Rotation of a ruagnetic pole round a current, and of 

a current round a magnetic pole. — A current and a magnetic 

pole neither attract nor repel 

each other, but tend to rotate 

about each other, the action 

being at right angles to the line 

joining them. Hence a mag- 
netic pole free to move will 

rotate round a current. This 

may be shown experimentally 

with apparatus like that shown 

in Fig. 434. The magnet NS 

is bent so that it may be pivoted 

on its middle point, the current 

being brought to an annular 

mercury cup, A, by means of a 

wire which dips into the mer- 
cury, leaving through the screw 

cup B. When a strong current is passing, the magnetic pole 

N rotates steadily, and by reversing the direction of the 

current the direction of rotation of the pole is reversed. 

Fm. 434. 



The rotation of a current round a pole may be shown by 
pivoting a wire bent in the form of an inverted letter U A 
(Fig. 435), on the top of a vertical 
horse-shoe magnet. The divided current 
passes through the mercury cups B and 
C, and leaves by the annular cups D and 
E which surround the magnet lower 
down. The cups B and C and the wires 
passing from them to the cups D and E 
are so pivoted upon the extremities of 
the magnet as to be free to rotate around 
its poles. When a strong current is 
passed through the circuit the wires will 
rotate in opposite directions round its 
two poles. 

The hypothetical currents that circulate round a magnetic 
molecule we shall call amperian currents, to distinguish them 
from the known current that traverses the solenoid. In 
strict accordance with this theory, the poles of the electro- 
magnet are determined by the direction of the current ^in the 
helix. The inductive influence of the electric current causes 
the amperian currents to take the same direction with itself, 
as represented in Fig. 436. 

By the amperian theory the earth's polarity is accounted 

Fig. 435. 

Fig. .436. 

for by assuming it to be girdled by electric currents, called 
earth currents, in planes approximately parallel to the equa- 
tor. A. person standing on the Arctic magnetic pgle of the 
earth would, if the currents were visible to him, see them 



(more properly their resultant, a current sheet) circulating 
round him towards his right from east to west, or in the same 
direction as the sun appears to him to go round the earth. 
According to Ampere's theory it is the tendency of the nearer 
portions of the earth currents and the amperian currents circu- 
lating round a magnetic needle to coincide in direction and to 
be parallel ; that causes the needle to point north and south. 
However well adapted this theory may be to explain most 
of the known phenomena of magnetism, it should be borne in 
mind that physicists of this generation value the theory 
rather as a help to the imagination and memory, than as a 
true statement of the facts. It is nearer the truth to say that 
the molecules are polarized as if currents were circulating 
around them ; of the actual existence of such currents we 
know nothing. So also of the real nature of polarity we 
know little or nothing. 

Section XVIII. 


515. Description of apparatus. 

Fig. 437. 

— A (Fig. 437) is a short coil 
of coarsB wire {i.e. the wire 
which it contains is com- 
paratively short), and has, 
of course, little resistance. 
B is along coil of fine wire 
having many turns. Coil 
A is in circuit with two 
Bunsen cells in multiple arc. 
This circuit we call the 
primary circuit, the current 
in this circuit the primary 
or inducing current, and the 
coil the primary coil. An- 



other circuit, having in it no battery or other means of gener- 
ating a current, contains coil B and a galvanoscope with an 
astatic needle.^ This circuit is called the secondary circuit, 
the coil the secondary coil, and the currents which circulate 
through this circuit are called secondary or induced currents. 

Experiment 1. — After all the connections are made, and a current is 
established in the primary circuit, and the galvanoscope needle is brought 
to zero, lower the primary coil quickly into the secondary coil, watching 
at the same time the needle of the galvanoscope to see whether it moves, 
and, if so, in what direction. Simultaneously with this movement there 

Fig. 438. 

is a movement of the needle, showing that a current must have passed 
through the secondary circuit. Let the primary coil rest within the 
secondary, until the needle comes to rest. After a few vibrations the 
needle settles at zero, showing that the secondary current was a tem- 
porary one. Now, watching the needle, quickly pull the primary coil 
out ; another deflection in the opposite direction occurs, showing that a 
current in the opposite direction is caused by withdrawing the coil. 

It is evident that in this case the current does not by its 
mere presence cause an induced current, but that a change in 
the relative positions of the two circuits, one of which bears 
a current, is necessary. 

1 This needle consists of two needles of about the same intensity with their poles 
reversed, fixed parallel with each other. Though the needles nearly neutralize each 
other and are therefore little affected by the field of the earth's magnetism, they are 
especially sensitive to the influence of the electric current properly situated. 


Experiment 2. — Place tlie primary coil within the secondary. Open 
the primary wire at some point and then close the circuit (Fig. 438) by 
bringing in contact the extremities of the wires. A deflection is pro- 
duced. As soon as the needle becomes quiet, break the circuit by sepa- 
rating the wires ; a deflection in the opposite direction occurs. 

The same phenomena occur when the primary current is 
by any means suddenly strengthened or weakened. 

An examination of the direction of tliese currents enables 
us to state the facts as follows : Starting a current in a 
primary, increasing the strength of the primary current, or 
moving the primary nearer while the current is steady, pro- 
duces a transitory current in the opposite direction in the 
secondary. Stopping the primary, diminishing the strength 
of the primary current, or moving the primary away while 
the current is kept steady, causes a transitory current in the 
same direction in the secondary. 

It is evident, therefore, that the conditions under which a 
current in the primary coil can cause a current in a neighbor- 
ing secondary depend upon some change either in the strength 
of the primary current or in the relative positions of the 
primary and secondary circuits. 

Experiment 3. — Introduce the bundle, D (Fig. 437), of soft iron wires, 
called the core^ into the primary coil, and make and break. the primary 
circuit as before. The deflections are now very much increased. 

Experiment 4. — Substitute a person for the galvanometer in the sec- 
ondary circuit, the person grasping some metallic handles made for the 
purpose and used as electrodes. The person experiences at the instant 
of making and breaking a peculiar sensation in his wrists and arms, 
called a shock. 

Experiment 5. — Introduce into the primary circuit the automatic 
make-and-break piece C (Fig. 437). Remove the core from the primary 
coil. Let a person grasp the electrodes of the secondary circuit. This 
person experiences a series of shocks which seem to him almost, if not 
quite, continuous. These shocks can be intensified to suit the pleasure 
of the person who is receiving them, by gradually lowering the core into 
the x^rimary coil. 


Experiment 6. — Eeflecting that you have hitherto found a coil of 
wire having a current passing through it acting as a magnet, you have 
now an opportunity to try the 
converse, i.e. to see whether a 
magnet may not take the place of 
a current-hearing coil. Introduce 
suddenly a bar-magnet (Fig. 439) 
into the secondary coil, as in Ex- 
periment 1 ; a deflection is pro- 
duced. "Withdraw it and an op- 
posite deflection occurs. 

The act by wliich the prim- 
avj, or a magnet, causes a cur- 
rent in a neighboring second- 
ary is called inagneto-electric 

516. Law governing E.M.F. of induced currents. — In any 
induced current the E.M.F. at any instant is jproportional to 
the rate of change in the number of lines of force passing through 
the circuit at that instant. If there be no change in the 
number of lines there is no induced E.M.F., however rapid 
the motion may be. 

Experiment 7. — Introduce a long bar magnet, NS (Fig. 440), into a 
short coil of wire, C, connected to a galvanometer, G. Place the coil 

half-way between the poles of 

Fig. 439. 


Fig. 440. 

the magnet *and move it rapidly 
(say) one centimeter toward either 
pole ; no movement of the gal- 
vanometer needle occurs although 
all the lines of force of the mag- 
net pass through C. 

Now place C near one of the 
ends of the magnet ; a similar mo- 
tion produces a large deflection. 

By means of an induction coil, a current of a few amperes 
circulating in the primary under an E.M.F. of not more than 



10 or 20 volts can be caused to yield currents in the secondary 
urged by an E.M.F. of many thousand volts. 

517. Faraday^ s law of induction. — If any conducting cir- 
cuit be placed in the magnetic field, then, if a change of 
relative position or change of strength of the primary current 
cause a change in the number of lines of force passing through 
the secondary, an electro-motive force is set up in the sec- 
ondary proportional to the rate at which the number of lines 
of force included by the secondary is varying. 

Consider the case of induction by a magnet. Let S (Fig. 
441) be a secondary circuit and IST a magnet projecting a cer- 
tain number of lines of force 
through the circuit. If S 
be moved nearer to the mag- 
net, say to S', a'much greater 
number of lines of force of 
the magnet pass through the 
circuit than when in its for- 
mer position, owing to the 
divergence of the lines as 
they recede from the pole. 
We may now understand, 
in part, the reason why a core of soft iron so greatly increases 
the induced current. It acts like a lens in focusing or con- 
centrating more lines of force from the magnetic inducer 
through the aperture of the secondary, and therefore any 
movement makes a greater rate of change, and hence a greater 
induced electro-motive force. 

518. Tlarth induction. — Call to mind that the earth itself 
is a great magnet, and that its lines of force pass through 
our atmosphere from pole to pole, and it will be easy to con- 
ceive that the mere motion of a coil of wire about an axis 
properly placed ^ is all that is necessary to produce a current. 

Fig. 441. 

1 The coil should he placed at right angle 
locality. Why ? 

to the direction of the dip at, the 



Such a coil with a galvanometer G in circuit is represented 
in Fig. 442. The rotation of the coil across the magnetic 

Fig. 442. 

lines of force of the earth effects a change of the number of 
lines of force passing through it, as may be understood by 
inspection of Fig. 443, and this creates temporary currents 
in the coil. 

By examination of Fig. 443 it will be seen that there are 
in each complete rotation of the coil two points (as A in the 
figure) where the coil encloses 
a maximum number of lines of 
force. In this position the in- 
duced current vanishes, for at 
this instant the number of 
lines is neither increasing nor 
diminishing. As the coil in 
its rotation approaches these 
points, the number of lines of 
force increases, and after leav- fig. 443. 


ing it the number diminishes. This will evidently cause a 
change in the direction of the induced current twice during 
each revolution. If then by means of a covimutator, a (Fig. 
452), the direction of the current in the galvanometer be 
changed relatively to its direction in the coil at each half 
revolution, we have an intermittent current constant in direc- 
tion through the galvanometer. 

519. Lenz^s law. — Recurring to the primary and secondary 
circuits we remark that the motion of the one or the other 
may be in arcs of circles or in any way, yet the motion may 
always be resolved so as to give a resultant indicating ap- 
proach or recession. The law by which the direction of the 
induced current is determined is known as Lenz's law, and 
may be expressed as follows: "7?^ all cases of induction the 
direction of the induced current is such as to oppose the motion 
which produces itJ^ Thus approach develops an opposite cur- 
rent, since opposite currents resist approach, while recession 
develops a current of similar direction, since similarly directed 
currents attract one another and thus resist recession. 

520.. Mechanical energy transformed into electric energy^ and 
vice versa. — It is, then, apparent that the current developed 
in the secondary circuit is at the expense of mechanical 
energy, and thus mechanical energy is transformed into 
electric energy.^ 

Eeturning to the apparatus (Figs. 434 and 435) in which 
we have the movement of a magnet-pole in the field of a 
current, and of a current in the field of a magnet, — if we 
replace the battery by a sensitive galvanometer, it is evident 
from the above discussion that on rotating the magnetic pole 
(Fig. 444) or the conductor (Fig. 445), in other words on 

1 The student might have been able to prophesy Lenz's law by reasoning thus : 
Suppose coil B approaches circuit A, we know (1) that electrical energy appears in 
B ; therefore from the doctrine of conservation of energy we know (2) that work 
must have been done ; hence, if work has been done, there must have been a repel- 
lent force between A and B. 



reversing the operations indicated in § 514, currents will 
traverse the circuits and their presence may be detected by 
the deflection of the needles of the galvanometers G. If we 
examine the direction of 
these induced currents, 
we shall discover that 
they are always opposite 
to the current which 
would actually cause the 
rotation. Since the in- 
duced current is opposite 
in direction to the cur- 
rent which would cause 
the motion, it is evident 
that the electro-magnetic ^^^- ^^- ^^^- ^^• 

effect of the induced current is to oppose the motion taking 
place in the field, in conformity with Lenz's law. 

We have seen that the same apparatus may be used either 
to transform electric into mechanical energy, or to transform 
mechanical into electric energy. 

521. Self-induction. — ''Extra currents.''^ — Not only does a 
current at starting and stopping or changing strength act on 
neighboring conductors, generating currents in them, but it 
acts upon itself by a process which is called self-induction. 
A current starting or increasing creates an oppositely directed 
current not only in its neighbor, but also in its own wire. A 
current does not start instantaneously ; it takes a certain 
time — usually very short — to rise to its full strength. In 
other words the circular lines of magnetic force round a 
straight current do not spring into existence instantaneously, 
but expand gradually like the widening ripples produced 
when a stone is dropped into still water. But when started 
it tends to persist, so that if its circuit be suddenly broken, 
it does not stop instantaneously. The lines of force gradually 


collapse, but tlie point of interest is that this, collapse 
gives rise to an electrical push, or E.M.F. far greater than 
that which maintained the current, and this sudden drive 
forward of electricity in the wire at the instant the circuit is 
broken causes the spark seen on breaking a circuit ; and the 
more sudden the break the more violent the spark. If a 
current pass through the helix of an electro-magnet, owing to 
the permeability of the iron a far larger number of lines of 
force traverse its circuit than if the core were removed ; and 
hence, at the stoppage of the current, a correspondingly 
greater impulse operates in the wire and creates a c